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FLOR1AN  CAJORI 


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HOPKINS  AND  UNDERWOOD'S  NEW 
ARITHMETICS 

ADVANCED  BOOK 


THE  MACMILLAN  COMPANY 

NEW  YORK   •    BOSTON  •    CHICAGO 
ATLANTA  •    SAN  FRANCISCO 

MACMILLAN  &  CO.,  LIMITED 

LONDON  •    BOMBAY  •    CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  LTD. 

TORONTO 


HOPKINS  AND  UNDERWOOD'S 

NEW    ARITHMETICS 


ADVANCED   BOOK 


BY 

JOHN  W.    HOPKINS 

SUPERINTENDENT   OF  THE   GALVESTON    PUBLIC   SCHOOLS 
AND 

P.   H.   UNDERWOOD 

TEACHER   OF  MATHEMATICS,    BALL   HIGH   SCHOOL,    GALVESTON,   TEXAS 


gorfe 
THE   MACMILLAN   COMPANY 

1908 

All  rights  reserved 


K7-3 


COPYRIGHT,  1903,  1907, 
BY  THE  MACMILLAN  COMPANY. 

Set  up  and  electrotyped.    Published  August,  1903. 
New  edition  revised.    Published  December,  1907. 

CAJOR1 


NorfoooU 

J.  8.  Gushing  Co.  —  Berwick  &  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


PREFACE 

THIS  book  assumes  a  working  knowledge  of  the  four 
fundamental  rules  as  applied  to  integers  and  United 
States  Money.  It  contains  the  essentials  of  practical 
arithmetic  arranged  by  topics  in  conformity  with  the 
courses  of  study  in  some  of  the  best  school  systems,  each 
chapter  representing  a  year's  work  commencing  with  the 
fifth  grade. 

It  aims  to  teach  principles  rather  than  rules.  As  the 
unitary  method  is  the  one  most  natural  to  the  young 
learner,  the  first  two  chapters  give  prominence  to  this 
style  of  reasoning.  Chapters  III  and  IV  give  a  thorough 
review  of  arithmetic  principles  and  practice.  In  these 
chapters  the  method  of  ratio  is  brought  into  prominence. 
Science  to  be  of  value  must  be  more  or  less  deductive  in 
form. 

Characteristic  features  of  the  book  are  the  early  intro- 
duction of  Decimals,  the  large  number  of  problems  based 
on  the  industrial  resources  of  our  country,  the  clearness 
and  directness  with  which  problems  are  illustrated,  and 
the  omission  of  complicated  problems  of  doubtful  utility. 

The  aim  of  teaching  arithmetic  is  culture,  accuracy, 
and  rapidity  in  the  computation  of  problems  arising  in 
actual  life,  and  the  acquisition  of  correct  methods  of 
reasoning.  This  aim  is  always  kept  in  view.  However, 
as  students  possess  the  power  of  learning  readily  to  work 
processes,  and,  furthermore,  as  the  practice  of  arithmetic 

918235 


vi  PREFACE 

is  of  more  importance  to  the  majority  of  people  than  the 
theory,  attention  is  paid  especially  to  the  art  of  compu- 
tation. 

The  book  contains  a  short  introduction  to  the  method 
of  obtaining  approximate  results  correct  to  any  required 
degree  of  accuracy.  This  matter  is  new,  but  it  is  believed 
that  it  is  well  worthy  of  consideration. 

Chapters  III  and  IV  contain  all  the  arithmetic  and  men- 
suration that  is  most  needful  to  be  known,  and,  in  fact, 
will  be  found  comprehensive  enough  to  suit  the  require- 
ments of  pupils  taking  a  survey  of  commercial  arithmetic. 

JOHN  W.  HOPKINS, 
P.   H.   UNDERWOOD. 
GALVESTON,  TEXAS, 

September  9, 1907. 


CONTENTS 

PAGE 

PREFACE v 

CHAPTER  I 

NUMERATION  AND  NOTATION 1 

DECIMALS 5 

ADDITION  AND  SUBTRACTION 7 

MULTIPLICATION  AND  DIVISION 11 

BILLS 22 

MEASURES  AND  MULTIPLES 24 

TESTS  OF  DIVISIBILITY 27 

GREATEST  COMMON  DIVISOR 30 

LEAST  COMMON  MULTIPLE 31 

FRACTIONS 33 

REDUCTION  OF  FRACTIONS 35 

ADDITION  OF  FRACTIONS 39 

SUBTRACTION  OF  FRACTIONS 41 

CANCELLATION 42 

MULTIPLICATION  OF  FRACTIONS 44 

DIVISION  OF  FRACTIONS.    RATIO 48 

DECIMALS 53 

MULTIPLICATION  AND  DIVISION  BY  POWERS  OF  TEN       .        .  54 

ADDITION 57 

SUBTRACTION 57 

MULTIPLICATION 59 

DIVISION o  61 

vii 


viii  CONTENTS 

PAGE 

REDUCTION  OF  FRACTIONS  TO  DECIMALS  AND  REDUCTION  OF 

DECIMALS  TO  FRACTIONS     .......  66 

MISCELLANEOUS  EXAMPLES 67 

COMPLEX  FRACTIONS 70 

AREAS  OF  RECTANGULAR  FIGURES 71 

COMPUTATION  ON  THE  BASIS  OF  100,  1000,  2000       ...  73 

PERCENTAGE 76 

INTEREST  AND  PROPERTY  INSURANCE        .        .        ...  79 

CHAPTER  II 

COMPOUND  QUANTITIES  . 81 

REDUCTION  OF  COMPOUND  QUANTITIES 82 

ADDITION  .  .  . 93 

SUBTRACTION  ...  .  .  .  .  .  .  .96 

MULTIPLICATION .  .99 

DIVISION  .  .  .  . 100 

EXPRESSION  OF  ONE  QUANTITY  AS  A  FRACTION  OF  ANOTHER 

QUANTITY  .  .  . 102 

MEASUREMENTS    .        ...        .        .        .        .        .        .  103 

VOLUMES  OF  RECTANGULAR  SOLIDS  .        .        .        .         .        .  105 

AREAS  OF  PARALLELOGRAMS,  TRIANGLES,  AND  TRAPEZOIDS  .  107 

BOARD  MEASUREMENT  ........  110 

MASONRY  AND  BRICKLAYING 112 

CARPETING .  .  113 

MISCELLANEOUS  EXERCISES 115 

REVIEW  OF  FRACTIONS 120 

PERCENTAGE .  .  .  122 

To  FIND  A  NUMBER  WHEN  A  PER  CENT  OF  IT  is  GIVEN  .  124 
TO  FIND  WHAT  PER  CENT  ONE  NUMBER  IS  OF  ANOTHER 

NUMBER .  .  .  .  127 

COMMERCIAL  DISCOUNTS 130 

PROFIT  AND  Loss 133 


CONTENTS  ix 

PAGE 

COMMISSION  AND  BROKERAGE     .    •    .    .     .    *    .        .        .        .  140 

INTEREST .        .  144 

SPECIFIC  GRAVITY        . 148 

RATIO    .        .        ...        .        ...        .        .        .  154 

PROPORTION .        .        .        .        .  156 

REVIEW          .        .        .        .   <     .   • 158 

CHAPTER  III 

GENERAL  REVIEW  BY  TOPICS 163 

ADDITION       . .        .  163 

SUBTRACTION         ....        .        .        .  168 

MULTIPLICATION 172 

PARTICULAR  SHORT  METHODS  OF  MULTIPLICATION         .        .  173 

DIVISION        .        . 177 

LONGITUDE  AND  TIME         .        .        ...        .        .        .  181 

STANDARD  TIME 188 

APPROXIMATIONS.         CONTRACTED      PROCESSES,       GENERAL 

METHODS  OF   SOLUTION       ....        .        .        .  190 ' 

LANGUAGE  OF  MATHEMATICS 196 

s\ 

'RATIO 200 

COMPOUND  PROPORTION        .        .        ....        .        .  205 

PARTNERSHIP         .        ....        .        .        .        .        .  208 

PERCENTAGE          .        . 210 

INTEREST .        .        .  217 

EXACT  INTEREST .  .  218 

INVERSE  QUESTIONS  IN  INTEREST      . 219 

REVIEW .222 

REVIEW  QUESTIONS      . 224 

PROMISSORY  NOTES .        .        .  225 

BANKERS'  INTEREST 228 

COMMERCIAL  DISCOUNT 231 

PARTIAL  PAYMENTS.     UNITED  STATES  RULE    .        .        .        .  232 


X  CONTENTS 

PAGK 

MERCHANTS*  RULE 234 

EXCHANGE 236 

VALUE  OF  FOREIGN  COINS  .        .        .        .    .     .        .        .        .  241 

ENGLISH  MONEY 242 

FOREIGN  EXCHANGE 244 

STOCKS  AND  BONDS 246 

CUSTOMS  AND  DUTIES 252 

CHAPTER  IV 

INVOLUTION 255 

EVOLUTION 259 

PROBLEMS  INVOLVING  SQUARE  ROOT         .....  264 

AREAS  OF  PLANE  TRIANGLES 267 

MENSURATION  OF  THE  CIRCLE 269 

SIMILAR  FIGURES 275 

SURFACES    OF    PRISM,    PYRAMID,     CYLINDER,     CONE,     AND 

SPHERE 277 

VOLUMES  OF  SOLIDS 280 

MEASURE  OF  TEMPERATURE 282 

METRIC  SYSTEM  OF  WEIGHTS  AND  MEASURES         .        .        .  284 

ANNUAL  INTEREST 291 

COMPOUND  INTEREST 292 

MISCELLANEOUS  TOPICS 295 

WORK  AND  TIME 295 

MOTION  IN  THE  SAME  OR  OPPOSITE  DIRECTIONS      .        .        .  298 

CLOCKS 299 

MISCELLANEOUS  EXAMPLES  BY  TOPICS  (A)      ....  303 

MISCELLANEOUS  EXAMPLES  (B) 318 

TABLES  .                                                                                           ,  325 


HOPKINS  AND  UNDERWOOD'S  NEW 
ARITHMETICS 

ADVANCED    BOOK 


ADVANCED   BOOK   OF   AEITHMETIC 

CHAPTER   I 

Arithmetic  is  the  science  of  numbers  and  the  art  of  com- 
putation. 

The  fundamental  operation  in  arithmetic  is  counting. 

The  result  of  counting  is  a  number. 

Any  one  of  the  natural  numbers,  one,  two,  three,  etc.,  is 
called  an  integer,  or  -whole  number. 

A  unit  is  one  thing,  or  a  group  of  things  regarded  as  a 
single  thing. 

NUMERATION 

The  number  names  are  one,  two,  three,  four,  five,  six, 
seven,  eight,  nine,  ten,  eleven,  twelve,  thirteen,  fourteen, 
fifteen,  sixteen,  seventeen,  eighteen,  nineteen,  twenty, 
thirty,  forty,  fifty,  sixty,  seventy,  eighty,  ninety,  hun- 
dred, thousand,  million,  billion,  trillion,  etc.  By  combin- 
ing these  number  names  all  numbers  may  be  expressed  in 
words. 

Ones,  tens,  hundreds,  thousands,  ten-thousands,  are  re- 
spectively called  units  of  the  first  order,  units  of  the 
second  order,  units  of  the  third  order,  units  of  the  fourth 
order,  units  of  the  fifth  order. 

In  our  system  of  naming  numbers  ten  units  of  any 
order  are  equal  to  one  unit  of  the  next  higher  order.  On 
this  account  our  system  is  known  as  the  decimal  system. 

Numeration  is  the  naming  of  numbers. 


2  ADVANCED  BOOK  OF  ARITHMETIC 

NOTATION 

Every  number  can  be  expressed  by  one  or  more  of  the 
following  figures,  sometimes  called  Arabic  numerals :  1,  2, 
3,  4,  5,  6,  7,  8,  9,  0.  The  first  nine  of  these  figures  are 
called  digits,  or  significant  figures. 

Tens  are  written  in  the  second  place ;  hundreds,  in  the 
third  place ;  thousands,  in  the  fourth  place ;  millions,  in 
the  seventh  place.  For  example,  four  thousand  seven 
hundred  eighty-nine  is  written  4,789.  This  number  may  be 
read  four  thousands  seven  hundreds  eight  tens  nine  ones. 
The  nine  ones  occupy  the  first  place ;  the  eight  tens,  the 
second  place ;  the  seven  hundreds,  the  third  place ;  the  four 
thousands,  the  fourth  place.  To  read  a  number  expressed 
by  more  than  three  figures,  begin  at  the  right,  that  is,  with 
units  of  the  first  order,  and  mark  off  with  commas  the 
figures  in  groups  of  three  each.  The  three  places,  or 
orders,  in  which  units  of  the  first  order  occur  constitute 
what  is  called  the  units'  period ;  the  next  three  places,  the 
thousands'  period  ;  the  next  three  places,  the  millions'  period; 
the  next  three,  the  billions'  period,  etc.  As  an  illustration 
take  the  number  1734902309 ;  marking  this  number  off 
into  periods,  it  becomes  1,734,902,309;  this  is  read  one 
billion  seven  hundred  thirty-four  million  nine  hundred 
two  thousand  three  hundred  nine.  Observe  that  each 
period  has  its  hundreds,  tens,  and  units.  The  periods 
most  used  are  the  units,  thousands,  and  millions.  The  bil- 
lions period  is  rarely  used.  The  billionth  part  of  a  great 
circle  of  the  earth  is  less  than  2  inches.  The  names  of  a 
few  of  the  succeeding  periods  are  trillions,  quadrillions,, 
quintillions,  and  sextillions. 

Notation  is  the  expression  of  numbers  by  means  of 
symbols. 


NOTATION  3 

EXERCISE  1 

Express  in  words : 

1.  45289.  10.    910003.  19.    8307308. 

2.  90208.  11.    728000.  20.    8000401. 

3.  75307.  12.   400098.  21.    7000014. 

4.  392394.  13.    902023.  22.    6100079. 

5.  738211.  14.    630006.  23.    3927173. 

6.  328993.  is.    1000000.  24.    5009020. 

7.  401012.  16.    2227001.  25.    8000904. 

8.  300287.  17.    3456000.  26.    6203003. 

9.  200020.  18.    9287003.  27.    9000090. 

28.  What  is  the  largest  whole  number  expressed  by 
two  figures? 

29.  What  is  the  smallest  whole  number  expressed  by 
two  figures? 

30.  What  is  the  largest  whole  number  expressed  by 
three  figures? 

31.  What  is  the  smallest  whole  number  expressed  by 
three  figures? 

32.  Write  the  largest  number  expressed  by  the  figures 
0,  4,  5. 

33.  Write  the  smallest  number  expressed  by  the  figures 
0,  4,  5. 

34.  Write  three  numbers  expressed  by  the  figures  2,  3, 4. 

35.  Write  four  numbers  expressed  by  the  figures  7,  6, 8. 

36.  What  is  the  largest  number  expressed  by  the  figures 
7,  3,  2,  8? 

37.  What  is  the   smallest   number   expressed   by  the 
figures  2,  5,  3,  4? 


4  ADVANCED  BOOK  OF  ARITHMETIC 

EXERCISE  2 

Write  in  figures : 

1.  Four  thousand,  eight  hundred  twenty-seven. 

2.  Nine  thousand,  seven  hundred  one. 

3.  Sixty-eight  thousand,  four  hundred  fifty-two. 

4.  Forty-seven  thousand,  three  hundred  eight. 

5.  Ninety  thousand,  six  hundred  four. 

6.  Eighty-seven  thousand,  one  hundred  one. 

7.  Twenty-two  thousand,  three  hundred  eleven. 

8.  Twelve  thousand,  fifteen. 

9.  Eighteen  thousand,  eighteen. 

10.  Fourteen  thousand,  thirty-four. 

11.  Thirteen  thousand,  five. 

12.  Ninety  thousand,  nine. 

13.  Fifty-four  thousand,  eleven. 

14.  Seventy-three  thousand,  one. 

15.  Six  hundred  four  thousand,  two  hundred  one. 

16.  One  hundred  sixty-three  thousand,  ten. 

17.  One  hundred  one  thousand,  three  hundred. 

18.  One  hundred  thousand,  seven. 

19.  Four  hundred  ten  thousand,  one  hundred  twenty, 
seven. 

20.  Five  hundred  four  thousand,  three  hundred  eight. 

21.  Five  hundred  thousand,  eleven. 

22.  Six  hundred  thousand,  seventeen. 

23.  Nine  hundred  ninety  thousand,  fifteen. 

24.  Two  hundred  one  thousand,  one. 

25.  Seventy-two  thousand,  four. 


READING   DECIMALS  5 

DECIMALS 

NOTE.  Pupils  should  draw  and  measure  distances  such  as  9.3  centi- 
meters, 2.87  inches. 

The  law  pervading  the  decimal  system  of  notation  is 
the  value  of  a  digit  in  any  place  is  always  ten  times  the 
value  of  the  same  digit  written  in  the  next  place  to  the 
right.  A  familiar  illustration  of  this  law  is  the  notation 
of  United  States  money.  For  example,  $5.55.  The  5  on 
the  left  is  ten  times  the  value  of  the  second  5,  and  the  second 
5  is  ten  times  the  value  of  the  5  on  the  right.  The  period 
separating  dollars  and  cents  is  called  the  decimal  point. 

The  first  place  to  the  right  of  the  decimal  point  is 
called  the  tenths'  place  ;  the  second  place,  the  hundredths' 
place  ;  the  third  place,  the  thousandths'  place  ;  the  fourth 
place,  the  ten-thousandths'  place  ;  the  fifth  place,  the  hun- 
dred-thousandths' place;  the  sixth  place,  the  million  ths' 

place. 

READING  DECIMALS 


Since  .47  =  4  tenths  7  hundredths  = 

Therefore,  .47  is  read  forty-seven  hundredths. 

Since  .372  =  3  tenths  7  hundredths  2  thousandths  =  ^ 
+  To  -o  +  To2o  o  =  iVoV  Therefore,  .372  is  read  372  thou- 
sandths. 

In  general  a  decimal  is  read  by  regarding  it  as  a  whole 
number  and  adding  the  name  of  the  place  the  right-hand 
digit  occupies.  In  reading  decimals  and  should  not  be 
used  except  to  connect  the  integral  and  decimal  parts  of 
the  number.  For  example,  500.005  is  read  five  hundred 
and  five  thousandths.  .505  is  read  five  hundred  five 
thousandths.  8.0379  is  read  eight  and  three  hundred 
seventy-nine  ten-thousandths.  .8379  is  read  eight  thou- 
sand three  hundred  seventy-nine  ten-thousandths. 


6  ADVANCED  BOOK  OF  ARITHMETIC 

EXERCISE  3 
Read : 

1.  6.2.  7.  6.201.  is.  5.0067. 

2.  7.9.  8.  4.027.  •      14.  7.0123. 

3.  8.4.  9.  5.029.  is.  9.1238. 

4.  4.32.  10.  9.001.  16.  .0003. 

5.  .12.  11.  6.034.  17.  .0054. 

6.  5.17.  12.  8.295.  is.  .4008. 

Write : 

1.  33  hundredths.  3.    329  thousandths. 

2.  2005  "thousandths.  4.    101  thousandths. 

5.  Two  hundred  three  thousandths. 

6.  Seven  hundred  and  four  thousandths. 

7.  Nine  hundred  three  thousandths. 

8.  Nine  hundred  and  three  thousandths. 

9.  Six  thousand  seven  hundred  ten-thousandths. 

10.  Six   thousand   seven   hundred   and   one   ten-thou- 
sandth. 

11.  Five  hundred  ninety  ten -thousandths. 

12.  Five  hundred  and  ninety  ten-thousandths. 

13.  Six  thousand  one  ten-thousandths. 

14.  Seven  hundred  ten-thousandths. 

15.  Seven  hundred  ten  thousandths. 

16.  Five  hundred  thousandths. 

17.  Five  hundred-thousandths. 

18.  Two  hundred  seven  hundred-thousandths. 

19.  Two  hundred  and  seven  hundred -thousandths. 

20.  Five   thousand   two   hundred   two  hundred-thou- 
sandths. 


ADDITION  AND  SUBTRACTION  7 

21.  Six  thousand  four  hundred-thousandths. 

22.  Three  thousand  ten  thousandths. 

23.  Three  thousand  one  ten-thousandths. 

24.  Five  hundred  seventeen  ten-thousandths. 

25.  One  hundred  eleven  hundred-thousandths. 

26.  Seventy-eight  ten-thousandths. 

ADDITION   AND   SUBTRACTION 

The  result  of  combining  two  or  more  numbers  into  a 
single  number  is  called  the  sum. 

Addition  is  the  process  of  finding  the  sum  of  two  or 
more  numbers. 

Only  numbers  of  the  same  kind  can  be  added. 

The  symbol  for  addition  is  +,  and  it  is  read  plus. 

The  symbol  =  is  read  equal,  or  equals  ;  thus,  5  +  8  = 
13  is  read  five  plus  eight  equal  thirteen. 

The  difference  between  two  numbers  is  the  excess  of 
one  over  the  other. 

Subtraction  is  the  process  of  finding  the  difference  be- 
tween two  numbers. 

The  number  subtracted  is  the  subtrahend.  The  num- 
ber from  which  the  subtrahend  is  taken  is  the  minuend. 

The  result  of  subtraction  is  called  the  remainder,  or 
difference. 

The  sign  of  subtraction  is  — ,  and  is  read  minus.  Thus, 
8  —  5  =  3  is  read  eight  minus  five  equals  three. 

ILLUSTRATIVE  EXAMPLE.     From  913  take  537. 
913         800  +  100  +  13  =  913 
537        500+    30+    7  =  537 
376        300  +    70  +    6  =  376 

7  from  13  leaves  6  ;  3  from  10  leaves  7  ;  5  from  8 
leaves  3.  This  is  the  usual  explanation  given  by  teachers. 


8 


ADVANCED  BOOK  OF  ARITHMETIC 


When  a  figure  in  the  subtrahend  cannot  be  taken  from 
the  corresponding  figure  in  the  minuend,  a  unit  of  the 
next  higher  order  in  the  minuend  is  changed  to  ten  units 
and  then  added  to  the  figure  in  the  minuend.  A  better 
way  of  subtracting  is  :  7  and  6  are  13.  Write  6,  carry 
1.  1  and  3  are  4;  4  and  7  are  11.  Write  7,  carry  1.  1 
and  5  are  6  ;  6  and  3  are  9.  Write  3. 


EXERCISE  4 


The  following  table  gives  the  number  of  children  of 
school  age,  number  enrolled,  average  daily  attendance, 
and  total  expenditures  for  the  public  schools  by  states  and 
territories  for  the  school  year  1904  : 


STATE 

OR 
TERRITORY 


NUMBER 

OP 
CHILDREN 


NUMBER 
ENROLLED 


AVERAGE 

DAILY 
ATTENDANCE 


TOTAL 
EXPEND- 
ITURES 


North  Atlantic  Division. 


Maine 

163,931 

131,176 

98,257 

$  2,080,109 

New  Hampshire 

91,847 

65,673 

48,673 

1,376,899 

Vermont 

81,358 

66,535 

48,845 

1,176,784 

Massachusetts 

673,690 

494,042 

391,771 

16,436,668 

Rhode  Island 

108,471 

70,843 

51,692 

1,804,762 

Connecticut 

223,174 

163,141 

123,317 

3,795,260 

New  York 

1,859,824 

1,300,065 

963,780 

43,750,277 

New  Jersey 

514,585 

352,203 

239,505 

8,838,515 

Pennsylvania 

1,782,740 

1,200,230 

900,234 

26,073,565 

South  Atlantic  Division. 


Delaware 

50,695 

36,895 

25,300 

453,670 

Maryland 

347,594 

209,978 

130,065 

2,755,288 

District  of  Columbia 

64,766 

49,789 

39,300 

1,576,354 

Virginia 

611,555 

375,601 

224,769 

2,137,365 

West  Virginia 

319,874 

244,040 

158,264 

2,531,655 

North  Carolina 

666,782 

491,838 

318,055 

2,075.566 

South  Carolina 

490,214 

292,115 

214,133 

1,191,963 

Georgia 

789,939 

502,014 

310,400 

2,240,247 

Florida 

180,501 

122,636 

83,631 

945,848 

ADDITION 


South  Central  Division. 

Kentucky 

700,272 

501,482 

309,836 

$2,662,863 

Tennessee 

678,782 

502,330 

344,882 

2,602,141 

Alabama 

652,518 

365,171 

240,000 

1,252,247 

Mississippi 

563,019 

403,647 

233,175 

•1,868,544 

Louisiana 

483,967 

208,737 

155,794 

1,551,232 

Texas 

1,128,934 

722,904 

461,938 

6,200,587 

Arkansas 

467,821 

339,542 

212,131 

1,729,879 

Oklahoma 

164,882 

152,886 

93,495 

1,359,624 

Indian  Territory 

162,641 

38,422 

23,053 

643,616 

North  Central   Division. 

Ohio 

1,151,007 

835,607 

618,495 

15,802,002 

Indiana 

732,172 

550,732 

416,047 

9,363,450 

Illinois 

1,428,613 

978,554 

783,563 

21,792,751 

Michigan 

684,369 

497,299 

388,092 

9,158,014 

Wisconsin 

658,474 

461,214 

288,300 

7,885,050 

Minnesota 

566,397 

423,663 

272,500 

8,073,323 

Iowa 

672,271 

545,940 

373,023 

10,696,693 

Missouri 

965,598 

731,410 

464,706 

9,878,198 

North  Dakota 

110,938 

95,224 

58,442 

2,316,346 

South  Dakota 

130,844 

106,822 

73,700 

2,239,135 

Nebraska 

321,822 

278,930 

180,771 

4,774,146 

Kansas 

455,943 

390,236 

270,878 

5,684,579 

Western  Division. 

Montana 

63,106 

44,881 

31,471 

1,236,253 

Wyoming 

24,960 

14,512 

9,650 

253,551 

Colorado 

145,799 

134,260 

95,117 

3,984,967 

New  Mexico 

64,094 

39,704 

29,582 

353,012 

Arizona 

35,365 

21,088 

13,022 

438,828 

Utah 

98,762 

75,662 

56,183 

1,657,234 

Nevada 

9,013 

7,319 

5,182 

257,501 

Idaho 

54,700 

54,480 

39,817 

1,001,394 

Washington 

147,302 

161,651 

110,774 

4,053,468 

Oregon 

118,977 

103,877 

72,464 

1,803,339 

California 

363,846 

299,038 

222,182 

9,401,465 

Find  the  totals  for  each  of  the  above  divisions.     Also 
find  the  total  in  each  instance  for  the  entire  country. 


10  ADVANCED   BOOK  OF  ARITHMETIC 

EXERCISE  5 

The  following  table  gives  the  gross  and  net  earnings  of 
the  principal  railroads  of  Texas  for  the  eleven  months 
ending  May  31,  1907  : 

RAILROAD  GROSS  EARNINGS         NET  EARNINGS 

1.  C.  R.  I.  and  G.  $3,082,314.24  $    895,096.73 

2.  F.  W.  and  D.  C.  4,094,588.28  1,345,055.55 

3.  G.  H.  and  S.  A.  11,130,030.96  2,536,237.69 

4.  F.  W.  and  R.  G.  1,075,725.13  342,223.14 

5.  H.  and  T.  C.  6,572,660.00  2,242,022.12 

6.  G.  C.  and  S.  F.  12,510,936.83  3,365,234.82 

7.  H.  E.  and  W.  T.  1,284,929.31  511,161.15 

8.  I.  and  G.  N.  8,204,579.38  2,079,764.06 

9.  M.  K.  and  T.  9,989,708.55  2,239,523.41 

10.  St.  L.  S.  W.  of  T.          3,936,613.38         506,646.38 

11.  S.  A.  and  A.  P.  3,518,565.80      1,517,563.03 

12.  T.  and  N.  O.  4,103,849.13         968,031.21 

13.  T.  and  P.  15,456,714.54      5,427,188.11 

14.  Texas  Central  1,149,069.36         489,109.71 
Find  the  operating  expenses  (difference  between  gross 

and  net  earnings)  of  each  of  the  above  roads. 

15.  The  increase  in  net  earnings  of  the  G.  C.  and  S.  F. 
road  for  the  eleven  months  ending  May  31, 1907,  over  that 
of  the  corresponding  months  of  the  preceding  year  was 
$1,361,052.87.       Find  the   net   earnings   for   the   eleven 
months  ending  May  31,  1906. 

16.  A   like   increase   for   the    I.  and    G.    N.  road  was 
$1,235,847.06.       Find    its   net   earnings   for   the    eleven 
months  ending  May,  1906. 


MULTIPLICATION  AND  DIVISION  11 

MULTIPLICATION   AND   DIVISION 

97  +  97  +  97  +  97  +  97  =  ?  If  these  numbers  are 
added,  the  sum  will  be  485.  In  examples  of  this  character 
the  usual  process  is  as  follows  : 

97       Five  7's  are  35.     Write  5,  carry  3.     Five  9's  are 
_5  45,  and  3  are  48.     The  result  is  485. 
485       This  short  method  of  adding  is  called  multiplication. 

Multiplication  is  a  short  method  of  addition  when  the 
numbers  to  be  added  are  all  the  same.  The  number  to  be 
repeatedly  added  is  the  multiplicand.  The  number  in- 
dicating how  often  the  multiplicand  is  to  be  taken  as  an 
addend  is  the  multiplier.  The  result  of  a  multiplication 
exercise  is  the  product.  The  multiplicand  and  multiplier 
are  factors  of  the  product.  Since  the  multiplier  denotes 
number  of  times,  it  must  always  be  a  pure  number,  or  an 
abstract  number.  The  multiplicand  may  be  either  a  con- 
crete or  an  abstract  number.  The  product  is  concrete  or  ab- 
stract according  as  the  multiplicand  is  concrete  or  abstract. 

The  sign  of  multiplication  is  x ,  and  is  read  multiplied 
by,  or  times.  Thus,  $  34  x  7  means  $  34  is  to  be  multi- 
plied by  7.  7  x  $  34  means  7  times  $ 34. 

Is  7x8  =  8x7?     Is  9x4  =  4x9? 

Is  7x3x8  =  3x8x7?     Is  9x6x5  =  6x9x5? 

The  order  in  -which  numbers  are  multiplied  is  immaterial. 

(a)  How  many  times  is  $4  contained  in  $24  ?  (5)  What 
is  the  sixth  part  of  $24  ? 

These  two  examples  illustrate  the  two  kinds  of  division; 
the  first  is  to  determine  the  number  of  times  one  number 
or  quantity  is  contained  in  another  number  or  quantity. 
This  is  often  called  measuring.  The  second  is  to  deter- 
mine a  part  of  a  number  or  quantity,  and  is  often  called 
parting,  or  dividing. 


12  ADVANCED  BOOK  OF  ARITHMETIC 

Division  is  the  process  of  determining  how  often  one  num- 
ber is  contained  in  another,  and  also  of  determining  any  given 
part  of  a  number.  The  first  number  is  the  divisor;  the 
second  is  the  dividend  ;  the  result  is  the  quotient. 

When  divisor  and  dividend  are  concrete  numbers,  the 
quotient  is  abstract.  When  the  dividend  is  a  concrete 
number,  and  the  divisor  an  abstract  number,  the  quotient 
is  a  concrete  number. 

The  signs  of  division  are  -*-,  and  a  horizontal  stroke, 
the  dividend  written  above,  the  divisor  below.  Thus, 
27  -5-  3,  and  -2JL  indicate  that  27  is  dividend  and  3,  divisor. 

ILLUSTRATIVE  EXAMPLES 

(a)  The  area  of  Massachusetts  is  8,040  square  miles, 
and  the  average  number  of  inhabitants  per  square  mile, 
for  the  year  1900,  was  348.9.  Find  its  population. 

348.9 

8040         One   square   mile  averages   348.9  inhab- 
139560     itants ;    therefore,    8,040  square  miles   will 
27912          have  8,040  times  348.9  inhabitants. 
2805156.0 

(5)  The  area  of  the  United  States,  exclusive  of  posses- 
sions, is  3,026,000  square  miles,  and  the  estimated  popu- 
lation for  the  year  ending  June  30,  1905,  was  83,143,000. 
Find  the  average  number  of  inhabitants  per  square  mile. 

27.4 

3026)83143.  Observe  the  area  is  3,026  thousands 

6052  of  square  miles  and   the   population  is 

22623  83,143  thousands  of  inhabitants.    Hence, 

21182  the  required  quotient  will  be  obtained 

14410  by  dividing  83,143  by  3,026. 


MULTIPLICATION  13 

EXERCISE  6 

1.  An  office  desk  costs  $25.     How  much  will  3  such 
desks  cost?  8  desks?  36  desks?  49  desks? 

2.  Eggs  sell  for  26  /  per  dozen.     Find  the  cost  of  8 
dozen  ;  18  dozen  ;  94  dozen. 

3.  There  are  5,280  feet  in  a  mile.     How  many  feet  are 
in  19  miles?  in  76  miles? 

4.  How  many  days  are  in  39  weeks? 

5.  A  contractor  pays  in  wages  $78  a  day.     How  much 
will  he  pay  in  78  days? 

6.  How  many  hours  are  in  89  days? 

7.  A  train  travels  at  the  rate  of  34  miles  an  hour. 
How  far  will  it  run  in  47  hours? 

8.  How   many  acres   are   in   a    ranch  containing   98 
sections  of  land?     (1  section  =  640  acres.) 

9.  A  degree  on  a  meridian  of  the  earth  is  about  69 
miles.     How  many  miles  are  in  17  degrees  ? 

10.  A  cubic  foot  of  rock  weighs  148  pounds.     How 
many  pounds  do  3,297  cubic  feet  weigh? 

11.  The  rent  of  a  dwelling  is  $28  per  month.     Find 
the  rent  for  3  years. 

12.  A  gallon  of  water  contains  231  cubic  inches.     How 
many  cubic  inches  are  in  368  gallons? 

13.  A  book  has  360  pages,  each  page  has  32  lines,  and 
each  line  averages  9  words.     How  many  words  are  in  the 
book? 

14.  A  carpenter  earns  $3.20  a  day.     At  this  rate,  how 
much  wages  will  he  receive  in  299  days? 

15.  A  brick  mason  earns  $  5.60  a  day.     How  much  will 
he  earn  in  310  days  ? 


14  ADVANCED  BOOK  OF  ARITHMETIC 

EXERCISE  7 

1.  A  city  block  is  100  yards  long  and  90  yards  wide. 
Find  its  area. 

2.  Find  the  area  of  a  square  whose  side  is  84  yards. 

3.  Find  the  area  of  a  square  having  320  rods  for  a  side. 

4.  Find  the  area  of  a  rectangle,  the  length  being  140 
yards  and  the  width  84  yards. 

5.  Find  the  area  of  a  rectangle  238  yards  long  and  96 
yards  wide. 

6.  A  farm,  rectangular  in  shape,  440  yards  long  and 
380  yards  wide,  contains  how  many  square  yards  ? 

7.  Find  the  area  of  a  rectangle  75  rods  long  and  63 
rods  wide. 

8.  Find  the  area  of  a  grass  plot  240  feet  by  84  feet. 

9.  A  sheet  of  paper  18  inches  long  and  14  inches  wide 
contains  how  many  square  inches  ? 

10.  A  township  is  6  miles  long  and  6  miles  wide.     How 
many  square  miles  does  it  contain  ? 

11.  A  county,  having  the  shape  of  a  rectangle,  is  24 
miles  long  and  18  miles  wide.     How  many  square  miles 
are  in  its  area  ? 

12.  A  street  is  1760  yards  long  and  23   yards  wide. 
How  many  square  yards  does  it  contain  ? 

13.  A  garden  is  50  yards  long  and  44  yards  wide.    How 
many  square  yards  are  in  its  area  ? 

14.  A  city  lot  is  43  feet  wide  and  124  feet  deep.     How 
many  square  feet  are  in  its  area  ? 

15.  A  window  is  60  inches  by  48  inches.     How  many 
square  inches  are  in  its  area  ? 

16.  A  yard  is  36  inches.     How  many  square  inches  are 
in  a  square  yard  ? 


MULTIPLICATION  15 

EXERCISE  8 
Find  the  product : 

1.  $  79.94  x    8.  12.  $  79.29  x    7. 

2.  $  32.20  x    7.  13.  |  29.97  x    8. 

3.  $  79.49  x    6.  14.  $179.38  x    6. 

4.  $128.29  x    7.  is.  1373.39  x    5. 

5.  $399.39  x    9.  16.  $799.94  x    8. 

6.  $454.59  x  12.  17.  $822.50  x    9. 

7.  $729.38  x  11.  18.  $998.78  x  11. 

8.  $237.38  x    9.  19.  $778.75  x  12. 

9.  $720.99  x    7.  20.  $732.75  x    9. 

10.  $285.68  x    8.  21.  $928.34  x    7. 

11.  $  51.33  x    7.  22.   $653.82  x    5. 

23.  When  shoes  sell  for  $3.90  a  pair,  how  much  will  24 
pairs  of  shoes  cost  ? 

24.  If  overcoats  sell  for  $7.98,  find  the  price  of  20 
overcoats. 

25.  Mackintoshes  sell   for  $6.95  apiece.     How  much 
will  27  mackintoshes  cost? 

26.  When  wheat  is  84^  per  bushel,  how  much  will  384 
bushels  bring  ? 

27.  Find  the  price  of  349  bushels  of  corn   at   56^  a 
bushel. 

28.  Cheese  costs  13^  per  pound.     Find  the  price  of  54 
pounds. 

29.  Find  the  cost  of  325  pounds  of  sugar  at  6?  per 
pound. 

30.  An  acre  of  land  is  worth  $60.75.     Find  the  value 
of  100  acres. 


16  ADVANCED  BOOK  OF  ARITHMETIC 

EXERCISE  9 

On  the  map  of  the  United  States  published  by  the 
General  Land  Office,  Department  of  the  Interior,  1  inch 
represents  37  miles.  On  this  map  the  distances  in  inches 
between  the  cities  named  are  given  below : 

1.  New  Orleans  to  Chicago  22.75. 

2.  Savannah  to  Indianapolis  16.6. 

3.  Mobile  to  Toledo  21.89. 

4.  Richmond  to  St.  Louis  19.2. 

5.  Washington  to  San  Antonio  37.88. 

6.  Boston  to  Jackson  34.6. 

7.  Atlanta  to  Des  Moines  20.4. 

8.  Newport  to  St.  Louis  27.9. 

9.  New  York  to  Lincoln  32.2. 

10.  Chicago  to  San  Francisco  50.8. 

11.  St.  Louis  to  Portland,  Oregon,  47.1. 

12.  Memphis  to  Seattle  51.2. 

Find  the  distance  in  miles  between  each  of  the  above- 
named  cities. 

Find  the  number  of  inhabitants  in  the  states  named: 

STATE  AREAS  IN  SQ.  Mi.  NUMBER  OF  INHABIT- 

ANTS PER  SQ.  Mi. 

13.  Georgia  58,980  37.6 

14.  Iowa  55,475  40.2 

15.  Illinois  56,000  86.1 

16.  Louisiana  45,420  30.4 

17.  Michigan  57,430  42.2 

18.  New  Jersey  7,525  250.0 

19.  Ohio  40,760  102.0 

20.  Pennsylvania  44,985  140.0 


DIVISION  17 

EXERCISE  10 
Divide  and  prove  your  answers  correct : 

1.  77,354  -;- 16.         9.     99,392  -  36.       17.  828,374  +  56. 

2.  79,358  -*- 18.       10.     59,738  +  35.      is.  528,739  ^  64. 

3.  97,854  H-  20.       11.     49,399  -  40.       19.  629,394  -*-  72. 

4.  92,738  ^  21.       12.     99,988  -f-  42.      20.  273,579  -5-  81. 

5.  100,000  +  24.       is.     68,698  -*-  48.       21.  179,246  -*-  84. 

6.  73,948  -5-  25.       14.  123,456  -^  49.      22.  739,264  -r-  90. 

7.  69,593-^23.      is.  876,543-^-50.      23.  543,293 -f- 56. 

8.  85,376  -v-  32.   16.  789,295  -r-  54.   24.  665,670  +  81. 

EXERCISE  11 

1.  When  sugar  sells  for  6  cents  per  pound,  how  many 
pounds  can  be  bought  for  84  cents  ? 

2.  If  a  boy  walks  at  the  rate  of  3  miles  per  hour,  how 
long  will  it  take  him  to  walk  87  miles  ? 

3.  In  a  peck  there  are  8  quarts.     How  many  pecks  are 
there  in  3000  quarts  ? 

4.  If  a  bicyclist  rides  9  miles  an  hour,  how  many  hours 
will  it  take  him  to  go  from  St.  Louis  to  Indianapolis,  a 
distance  of  265  miles?     After  riding  19  hours,  how  far 
from  Indianapolis  will  he  be  ? 

5.  When  coal  costs  9  dollars  a  ton,  how  many  tons  can 
be  bought  for  3456  dollars  ? 

6.  A  teacher  receives  a  salary  at  the  rate  of  4  dollars  a 
day  for  every  day  he  teaches.     His  yearly  salary  is  716 
dollars.     How  many  days  are  in  the  school  year  ? 

7.  A  brick  mason  receives  4  dollars  a  day  for  every  day 
he  works.     How  many  days  must  he  work  to  earn  900 
dollars  ? 


18  ADVANCED  BOOK  OF  ARITHMETIC 

8.  How  many  feet  are  there  in  2,500  inches  ? 

9.  How  many  weeks  are  there  in  364  days  ? 

10.  If  a  dozen  penknives  cost  9  dollars,  how  many  dozen 
penknives  can  be  bought  for  126  dollars  ? 

11.  Plows  cost  12  dollars  a  piece.     How  many  can  be 
bought  for  192  dollars  ? 

12.  A  box  of  oranges  is  worth  3  dollars.     How  many 
such  boxes  can  be  bought  for  111  dollars  ? 

13.  When  a  barrel  of  pork  sells  for  12  dollars,  how 
many  barrels  must  be  sold  to  realize  5004  dollars  ? 

14.  A  section  foreman  rides  on  a  velocipede  at  the  rate 
of  11  miles  an  hour.     How  long  will  it  take  him  to  go 
from  Cincinnati  to  Cleveland,  a  distance  of  264  miles  ? 

15.  Divide  1000  dollars  among  8  persons,  giving  to  each 
the  same  sum  of  money. 

16.  A  flock  of  sheep  sells  for  966  dollars.     How  many 
sheep  are  in  the  flock,  if  each  sheep  sells  for  6  dollars? 

17.  A  man  has  795  dollars  in  5-dollar  gold  pieces.     How 
many  coins  has  he  ? 

18.  Hogs  sell  for  8  dollars  apiece.     At  this  price  how 
many  can  be  purchased  for  360  dollars  ? 

19.  A  box  of  soap  is  listed  at  4  dollars.     How  many 
such  boxes  can  be  purchased  for  980  dollars  ? 

20.  How  many  barrels  of  flour  can  be  bought  for  1002 
dollars,  when  flour  sells  for  6  dollars  a  barrel  ? 

21.  Oyster  crackers  cost  5  cents  a  pound.     How  many 
pounds  can  be  bought  for  95  cents  ? 

22.  By  buying  horses  at   75  dollars  each  and  selling 
them  at  84  dollars  each,  a  jobber  makes  a  profit  of  324 
dollars.     How  many  does  he  sell  ? 


DIVISION  19 

EXERCISE  12 

1.  How  many  bags  of  Rio  coffee  can  be  bought  for 
882  dollars,  if  one  bag  costs  21  dollars  ? 

2.  Currants  sell  for  14  dollars  a  barrel ;  at  this  price 
how  many  barrels  can  be  bought  for  546  dollars  ? 

3.  Granulated   sugar   is    worth   16    dollars   a   barrel. 
How  many  barrels  must  be  sold  to  realize  1264  dollars  ? 

4.  There  are  36  inches  in  one  yard.     How  many  yards 
are  in  100,000  inches  ? 

5.  There  are   32  quarts  in  one  bushel.     How   many 
bushels  are  in  7712  quarts  ? 

6.  How  many  days  are  in  3000  hours  ? 

7.  A  degree  on  a  meridian  of  the  earth's  surface  is  69 
miles  long.     Two  places  on  the  same  meridian  are  2484 
miles  apart.     How  many  degrees  apart  are  they  ? 

8.  How  many  square  yards  are  in  3276  square  feet  ? 

9.  A  gallon  contains  231  cubic  inches.     How  many 
gallons  are  in  a  barrel  containing  8316  cubic  inches  ? 

10.  Oolong  tea  costs  15  dollars  a  chest.     How  many 
chests  can  be  purchased  for  495  dollars  ? 

11.  A  barrel  of  sugar  weighs  325  pounds.     How  many 
barrels  are  in  105,625  pounds  ? 

12.  How  long  will  it  take  a  train,  rate  30  miles  an  hour, 
to  go  from  New  York  to  San  Francisco,  a  distance  of  3270 
miles,  if  5  hours  are  allowed  for  stops  ? 

13.  There  are  10  square  chains  in  an  acre.     How  many 
acres  are  in  10,000  square  chains  ? 

14.  Rhode  Island  contains  in  round  numbers  800,000 
acres.    Find  its  area  in  square  miles.   (640  acres  =  1  square 
mile.) 


20  ADVANCED  BOOK  OF  ARITHMETIC 

EXERCISE   13 

1.  If  land  is  worth  $68  an  acre,  how  many  acres  can 
be  bought  for  $4,624? 

2.  A  tract  of  land  is  sold  for  $4,795.50  at  the  rate  of 
$69.50  an  acre.     How  many  acres  are  in  the  tract  ? 

3.  When  horses  sell  for  $85.40  apiece,  how  many  can 
be  bought  for  $36,465.80  ? 

4.  If  the  price  of  wheat  is  76^  per  bushel,  how  many 
bushels  can  be  bought  for  $4,043.20  ? 

5.  When  cans  of  asparagus  sell  for  35^  each,  how  many 
can  be  bought  for  $85.75  ? 

6.  If  a  pair  of  patent  leather  shoes  sells  for  $3.85,  how 
many  pairs  must  be  sold  to  bring  $1,482.25  ? 

7.  A  clothier  invests  in  men's  trousers  $392. 04.     How 
many  does  he  buy,  supposing  each  pair  to  cost  $1.98  ? 

8.  If  a  keg  of  pickles  cost  $1.70,  how  many  kegs  can 
be  purchased  for  $28.90  ? 

9.  Chipped  beef  is  bought  at  17^  a  pound.     At  this 
rate,  how  many  pounds  can  be  bought  for  $361.25? 

10.  A  12-pound  sack  of  flour   retails   at  45^.      How 
many  sacks  can  be  bought  for  $322.65? 

11.  When  a  can  of  sardines  retails  for  27^,  how  many 
cans  will  $218. 70  buy? 

12.  A  farmer  gets  for  his  apples  $816.35  at  the  rate  of 
$1.45  per  barrel.     How  many  barrels  does  he  sell  ? 

13.  Corn  is  worth  45^  per  bushel.     How  many  bushels 
can  be  bought  for  $7,876.35  ? 

14.  Oats  are  worth  36  ^  per  bushel.     How  many  bushels 
can  be  bought  for  $142.56  ? 

15.  A   share    of   railway   stock   is   quoted  at   $78.50. 
How  many  shares  must  be  sold  to  realize  $3,061.50  ? 


DIVISION 


21 


EXERCISE    14 

The  following  table  gives  the  area  and  population  of 
some  of  the  principal  countries : 


COUNTRY 

1.  Austria-Hungary 

2.  Belgium 

3.  Denmark 

4.  France 

5.  German  Empire 

6.  Italy 

7.  Japan 

8.  Netherlands 

9.  Russia 

10.  Spain 

11.  United  Kingdom 


AREA  IN  SQ.  Mi. 

241,300 

11,370 

15,360 

207,050 

208,800 

110,600 

147,700 

12,560 

8,660,000 

194,800 

121,370 


POPULATION 

47,355,000 

7,161,000 

2,574,000 

39,300,000 

60,478,000 

33,604,000 

47,975,000 

5,592,000 

141,000,000 

18,618,000 

43,221,000 


Find  the  population  per  square  mile  of  each  of  the  above 
countries. 

12.    Reduce    1  to  a  decimal. 


17.00000 


2.83333 


.40476 
Reduce  to  decimals  : 


42  can  be  resolved  into  two  factors,  6 
and  7.  The  simplest  way  of  dividing  by 
42  is  to  divide  by  the  factors  6  and  7. 


;  H; 


rife; 


"•  M;  H;  i 

15-  ft;  W;  A;  IJ;  iflh 

Reduce  to  decimals,  correct  to  four  figures : 

16-    i  5  T  ;  i ;  T\  ;  iV  >  yV  5  i1!  5  iV  5  iV- 
17*     25Y  ;   M  ;   A  5   FT  5    II  5   yV' 


22 


ADVANCED  BOOK  OF  ARITHMETIC 


SPECIMEN  BILL 

GALVESTON,  TEXAS,  Jan.  31,  1907. 

MR.  A.  B.  C. 

In  account  with  K.  M.  &  CO., 

DEALERS    IN 

FURNITURE,  CARPETS,  RUGS,  &c. 


Jan. 

2 

3  Chairs  @  $2.25 

$6 

75 

10 

1  Library  Table 

25 

00 

15 

3  Rugs  @  $6.75 

20 

25 

20 

40  yd.  Matting  @  45  $ 

18 

00 

24 

2  Wardrobes  @  $17.50 

35 

00 

$105 

00 

Paid 

Feb.  1,  1903. 

K.  M.  &  CO. 

Per  M. 

\ 

MR.  P.  Q.  R. 


DALLAS,  TEXAS,  Feb.  5,  1907. 

Bought  of  M.  R.  S., 

RETAIL  GROCERS. 


Jan. 

5 

3  Ib.  Tea  @50$ 

$1 

50 

a 

28  Ib.  Sugar  @5\$ 

1 

4^ 

it 

3pks.  Potatoes  @  40$ 

1 

20 

(( 

7  Ib.  Bacon  @  15$ 

1 

05 

10 

8  Ib.  Butter  @  35$ 

•2 

80 

12 

3  cans  Salmon  @  17  $ 

51 

13 

6  Ib.  Sausage  @  12$ 

72 

®  o 

$>y 

25 

Paid 

Feb.  8. 

M.  E.  S. 

Per  X. 

DECIMALS  23 

EXERCISE  15 

Make  out  the  following  bills  and  receipt  them  : 

1.  Mr.  John  Rye  bought  of  William  Merchant, 

12  yd.  Calico  ......     @  9  1 

15  yd.  Sheeting    .....     @  7  ^ 

11  yd.  Flannel      .....     @          35  f 

2  Hats       .......     @    13.75 

18  yd.  Carpet       .....     @          75  1 

3  Smyrna  Rugs  .....     @  110.50 

2.  Mr.  VJ.  Hill  bought  of  F.  Warner  &  Co., 

5  Stoves    .......  @  $6.50 

3  doz.  Knives     .....  @  14.80 

2  Saws      .......  @  $1.50 

5  Iron  Beds  ......  @  115.75 

6  Wrenches  ......  @  $1.25 

3.  H.  Van  Oppen  bought  of  Hegel  &  Co., 

2  bu.  Potatoes    .....  @  $1.50 

5  Ib.  Tea  .......  @          75  ^ 

2  boxes  Herring      .     .     .     .  @  $1.95 

25  Ib.  Ham      ......  @          15  f 

45  Ib.  Sugar    ......  @ 


4.    Mr.  James  Kay  bought  of  Simpson,  Perdue  &  Co., 

50  Ib.  Sugar    ......  @  4%f 

15  cans  Tomatoes                      .  @  13  ^ 

27  cans  Corn  ......  @  16  ^ 

10  packages  Breakfast  Food   .  @  12J  ^ 

8  cans  Salmon    .....  @  18  ^ 

5  gal.  Maple  Sirup       .     .     .  @  $1.30 

25  Ib.  Butter  ......  @  37^ 

61b.  Y.  H.  Tea.     .     .     .     .  ©  75^ 


24  ADVANCED  BOOK  OF  ARITHMETIC 

MEASURES  AND  MULTIPLES 

NOTE.  Measures  and  Multiples  in  this  book  have  reference  only  to 
numbers  which  are  both  integral  and  abstract. 

A  number  is  prime,  or  is  said  to  be  a  prime  number, 
when  it  is  exactly  divisible  by  only  itself  and  unity. 

Thus,  1,  2,  3,  5,  7,  11,  13,  17,  19,  23,  29,  etc.,  are  prime 
numbers,  or  prime  factors. 

A  number  which  is  exactly  divisible  by  other  numbers, 
as  well  as  by  itself  and  unity,  is  called  &  composite 
number. 

Thus,  6,  8,  9,  10,  12,  14,  15,  16,  18,  20,  etc.,  are  compos- 
ite  numbers. 

An  even  number  is  one  which  is  exactly  divisible  by  2. 

Thus,  4,  6,  8,  10,  12,  etc.,  are  even  numbers. 

An  odd  number  is  one  which  is  not  exactly  divisible 
by  2. 

Thus,  3,  5,  7,  9,  11,  etc.,  are  odd  numbers. 

One  number  is  said  to  be  a  measure  of  another  number 
when  it  is  contained  an  exact  number  of  times  in  that 
other  number. 

Instead  of  the  word  "  measure,"  factor,  divisor,  and  sub- 
multiple  are  often  used.  Example:  5  is  a  measure,  factor, 
divisor,  or  submultiple  of  10,  15,  20,  25,  etc. 

By  Greatest  Common  Measure  of  two  or  more  numbers 
is  understood  the  greatest  measure  which  these  numbers 
have  in  common. 

Thus,  6  is  the  greatest  common  measure  of  12,  18,  30. 

Greatest  Common  Measure  is  usually  designated  by  the 
letters  G.  C.  M.  It  is  called  also  Greatest  Common  Divisor 
(G.  C.D.). 

If  a  number  measures  each  of  two  or  more  numbers,  it 
is  said  to  be  a  common  measure  of  those  numbers. 


SURES  AND  MULTIPLES 


25 

Thus,  2,  3,  and  6  are  common  measures  of  12,  18,  30. 

Two  or  more  numbers  are  prime  to  each  other  when  they 
have  no  common  measure  but  1. 

Thus,  8  and  9  are  prime  to  each  other;  so  also  are  15 
and  28  prime  to  each  other.  These  numbers,  while  prime 
to  each  other,  are  not  themselves  prime  numbers. 

The  result  obtained  by  multiplying  a  number  by  an 
integer  is  called  a  multiple  of  the  number. 

Thus,  the  multiples  of  8  are  8,  16,  24,  32,  40,  48,  etc. 

The  Least  Common  Multiple  of  two  or  more  numbers  is 
the  least  number  which  is  a  multiple  of  each  of  the  num- 
bers. In  other  words,  the  Least  Common  Multiple  of  two 
or  more  numbers  is  the  least  number  which  is  exactly 
divisible  by  each  of  the  two  or  more  numbers. 

Least  Common  Multiple  is  denoted  by  L.  C.  M. 

Example  1.     What  is  the  L.  C.  M.  of  8  and  12  ? 

Writing  the  multiples  of  8  and  12,  we  have: 

8        16        24        32        40        48        64        72         80 
12        24        36        48        60         72         84 

Notice  that  24  is  a  common  multiple  of  8  and  12.  So 
also  are  48  and  72  common  multiples.  The  L.  C.  M.  is  24. 

Example  2.  What  is  the  L.  C.  M.  of  12  and  18  ?  Here 
the  second  multiple  of  18  contains  12  as  a  factor.  Hence 
2  times  18  is  the  L.  C.  M. 

Example  3.  A  man  buys  two  kinds  of  sugar  dorie  up 
in  4-pound  bags  and  in  5-pound  bags.  What  is  the  least 
number  of  pounds  he  can  buy  so  as  to  have  the  same 
number  of  pounds  of  each  kind  ? 

Here  the  answer  is  obviously  a  multiple  of  4  and  5. 
The  L.  C.  M.  of  4  and  5  is  20.  Hence  he  buys  20  pounds 
of  each  kind. 


26  ADVANCED   BOOK  OF  ARITHMETIC 

EXERCISE   16 

1.  A  person  has  equal  sums  of  money  in  dimes  and  in 
25-cent  pieces.     Find  the  least  amount  he  can  have. 

2.  Fence  posts  in  two  fences  are  respectively  14  feet 
and  21  feet  apart.     What  is  the  smallest  distance  corre- 
sponding to  an  exact  number  of  feet  in  both  fences  ? 

3.  A  man  earns  $4  a  day.     How  many  days  must  he 
work  so  as  to  be  paid  in  10-dollar  notes  ? 

4.  If  a  person  earns  $6  a  day,  how  many  days  must 
he  work  to  be  paid  in  $  20  bills  ? 

5.  A  housewife  puts  her  flour  into  10-pound  and  6- 
pound  sacks,  and  has  the  same  quantity  in  the  10-pound 
sacks  as  she  has  in  the  6-pound  sacks.     What  is  the  least 
quantity  of  flour  she  can  have  ? 

6.  A  man   buys  two  grades  of   sheep  at  $4  and  $6 
a  head    respectively.      He  spends   the   same  amount    in 
the  purchase  of  the  two  grades  of  sheep.     What  is  the 
smallest  amount  he  can  spend  ?     How  many  sheep  can  he 
buy? 

7.  A  boy  buys  oranges  at  3^,  4^,  5^  apiece.   He  spends 
the  same  amount  on  each  kind  of  oranges.     What  is  the 
least  amount  he  can  spend  on  each  kind  ?     How  many 
oranges  does  he  buy  ? 

8.  A  person  spends  the  same  amount  of  money   on 
eggs  at  15^  a  dozen  and  at  20^  a  dozen.     What  is  the 
smallest  amount  he  can  spend  on  each  kind  ? 

9.  Three  bells  toll  at  intervals  of  4,  5,  and  6  seconds 
respectively.     If  they  start  at  the  same  time,  after  how 
many  seconds  will  they  toll  again  at  the  same  instant  ? 

10.    Find  in  feet  and  inches  the  least  distance  that  will 
be  measured  exactly  by  a  15-inch  and  an  18-inch  rule.     . 


TESTS  OF   DIVISIBILITY  27 

TESTS  OF  DIVISIBILITY 

A  number  is  exactly  divisible  by  2  when  its  units' 
figure  is  exactly  divisible  by  2.  Thus,  196  is  exactly 
divisible  by  2  since  6  is  divisible  by  2. 

A  number  is  exactly  divisible  by  5  if  its  units'  figure 
is  5  or  0. 

A  number  is  exactly  divisible  by  3  when  the  sum  of  its 
digits  is  exactly  divisible  by  3.  Thus,  735  is  exactly 
divisible  by  3  since  the  sum  of  its  digits,  15,  is  exactly 
divisible  by  3. 

A  number  is  exactly  divisible  by  6  when  it  is  even  and 
the  sum  of  its  digits  is  divisible  by  3.  Thus,  624  is  ex- 
actly divisible  by  6  since  it  is  an  even  number  and  the 
sum  of  its  digits,  12,  is  a  multiple  of  3. 

A  number  is  exactly  divisible  by  9  when  the  sum  of 
its  digits  is  exactly  divisible  by  9.  Thus,  765  is  exactly 
divisible  by  9  because  the  sum  of  its  digits,  18,  is  a  multi- 
ple of  9. 

A  number  is  exactly  divisible  by  11  when  the  differ- 
ence between  the  sums  of  its  digits  in  the  even  and  odd 
places  is  0  or  a  multiple  of  11.  Thus,  94,853,  is  exactly 
divisible  by  11  since  the  difference  between  the  sums  3  4- 
8  +  9  and  5  -h  4  is  a  multiple  of  11. 

A  number  is  exactly  divisible  by  25  when  the  number 
formed  by  its  two  right-hand  digits  is  exactly  divisible 
by  25.  Thus,  1,275  is  exactly  divisible  by  25  because  75 
is  exactly  divisible  by  25. 

A  number  is  exactly  divisible  by  8  when  the  number 
formed  by  its  three  right-hand  digits  is  exactly  divisible 
by  8.  Thus,  19,256  is  exactly  divisible  by  8  because  256 
is  divisible  by  8. 

The  same  rule  holds  for  125, 


28  ADVANCED  BOOK  OF  ARITHMETIC 

NOTATION 

22  is  a  short  way  of  writing  2x2. 

23  is  a  short  way  of  writing  2x2x2. 

24  is  a  short  way  of  writing  2x2x2x2. 

25  is  a  short  way  of  writing  2x2x2x2x2. 

The  result  of  taking  a  number  any  number  of  times  as 
factor  is  called  a  power  of  the  number.  Thus,  74  =  7  x  7  x 
7x7=2,401. 

2,401  is  the  4th  power  of  7. 

The  4  written  to  the  right  of  7  and  slightly  above  it  is 
called  the  index  or  exponent  of  the  power. 

Example  l.  Resolve  1,001  into  its  prime  factors.  1,001 
*s  no^  Divisible  by  2  because  its  units'  figure  is 
not  exactly  divisible  by  2.  It  is  not  divisible  by 
3  because  the  sum  of  the  digits,  1,  1,  is  not 
divisible  by  3.  It  is  not  divisible  by  5  because 
its  units'  figure  is  not  0  or  5.  7  is  contained  in  1,001,  143 
times.  143  is  divisible  by  11  because  the  sum  of  the 
digits,  1,  3,  equals  4,  the  digit  in  the  even  place.  Hence 
the  prime  factors  of  1,001  are  7,  11,  13.  Hence,  1,001  = 
7  x  11  x  13. 

Example  2.  Resolve  5,040  into  its  prime  factors,  and 
express  5,040  as  the  product  of  prime  numbers. 


5040 


2520 


Divide  by  2  as  often  as  possible.     Since  315 
1250     ends  in  5,  5  is  a  factor  of  315.     Divide  next  by 
3  as  often  as  possible. 

The  prime  factors  of  5,040  are  2,  2,  2,  2, 3,  3, 5,  7, 
5040  =  2x2x2x2x3x3x5x7 
=  24  x  32  x  5  x  7. 


630 


63 


- 

PRIME  FACTORS  29 


EXERCISE  17 

Resolve  into  prime  factors  and  express  each  number  as 
the  product  of  its  prime  factors : 

1.  8,  12,  16,  18,  20,  24,  27,  28,  30,  32,  36,  39,  40,  42. 

2.  45,  48,  49,  50,  56,  60,  65,  69,  72,  75,  77,  80,  84,  88, 
92. 

3.  98,  99,  111,  117,  119,  120,  124,  128,  132,  133,  135, 
140, 144. 

4.  240,  720,  343,  512,  216,  729,  736,  608,  544. 

5.  1,331,  11,011, 1,309,  858,  1,274,  891,  3,575. 

6.  Write    all   the  measures  of  each  of  the  following 
numbers:  36,  360,  200,  567,  576,  448. 

7.  Write  all  the  common   measures   of:   (a)    36,    24  ; 
(6)18,27;   (c?)48,72;   (d)21,63;   0)  32,  96;   (/)  18,  72. 

When  several  numbers  are  to  be  taken  as  a  whole  and 
made  the  subject  of  an  operation,  they  are  inclosed  in  a 
sign,  or  symbol,  known  as  a  parenthesis,  (  ).  Thus,  3  +  (9 
—  2)  signifies  that  3  is  to  be  added  to  the  difference  of  9 
and  2.  A  number  written  immediately  to  the  left  of  a 
parenthesis  denotes  multiplication.  Thus,  7x4+  3(8 
+  5)  means  7  times  4  is  to  be  added  to  3  times  the  sum 
of  8  and  5. 

A  composite  number  can  be  resolved  into  only  one  set 
of  prime  factors.  Thus,  the  prime  factors  of  36  are  2,  2, 
3,  3.  36  =  22  x  32.  The  product  of  no  other  prime  num- 
bers will  give  36. 

If  a  number  is  prime  to  each  of  two  other  numbers,  it  is 
prime  to  their  product. 

ILLUSTRATION.  If  7  is  prime  to  207  and  to  8,  then  7  is 
prime  to  8  x  207.  For  7  does  not  appear  among  the 
prime  factors  of  the  product. 


30  ADVANCED  BOOK  OF  ARITHMETIC 

Example  1.    Find  the  G.  C.  M.  of  48,  120,  168. 
Expressing  these  numbers  as  products  of  their  prime 
factors, 

48  =  24  x  3. 
120  =  23  x  3  x  5. 
168  =  23  x  3  x  7. 

2  is  contained  3  times  as  a  factor  in  168,  3  times  as  a  fac- 
tor in  120,  and  4  times  as  a  factor  in  48 ;  3  is  contained 
once  as  a  factor  in  each  of  the  numbers.  Hence,  the 
G.  C.  M.  =23X3  =  24. 

To  find  the  G.  C.  M.  of  two  or  more  numbers,  express 
each  of  the  numbers  as  the  product  of  its  prime  factors, 
then  take  the  product  of  the  prime  factors  common  to  all 
the  numbers,  each  factor  being  taken  the  least  number  of 
times  it  occurs  in  any  of  the  numbers. 


EXERCISE  18 

Find  the  G.  C.  M.  of: 

1.  16,  24.  12.  26,  117.  23.  64,  80,  96. 

2.  24,  32.  is.  57,  76.  24.  63,  84,  105. 

3.  18,  27.  14.  115,  161.  25.  64,  96,  224. 

4.  24,  36.  15.  144,  264.  26.  72,  108,  180. 

5.  45,  60.  16.  140,  252.  27.  88,  132,  220. 

6.  75,  90.  17.  20,  30,  40.  28.  126,  189,  252. 

7.  54,  72.  is.  30,  75,  105.  29.  144,  240,  336. 

8.  108,  180.  19.  36,  60,  84.  so.  162,  270,  378. 

9.  84,  96.  20.  39,  65,  91.  31.  168,  224,  392. 

10.  120,  156.  21.  60,  84,  132.  32.  252,  420,  588. 

11.  91, 105.  22.  54,  90,  108.  33.  264,  360,  600. 


LEAST  COMMON   MULTIPLE  31 

LEAST   COMMON  MULTIPLE 

Since  the  L.  C.  M.  of  two  or  more  numbers  is  exactly 
divisible  by  each  of  the  numbers,  it  follows  that  the 
L.  C.  M.  contains  all  the  prime  factors  of  each  of  the  given 
numbers. 

This  fact  suggests  a  method  of  finding  the  L.  C.  M.  of 
two  or  more  numbers. 

Example.     Find  the  L.  C.  M.  of  48,  60,  72. 
48  =  24  x  3. 
60  =  22x3x5. 
72  =  23  x  32. 

Any  multiple  of  48  must  contain  2,  4  times  as  a  factor. 
Any  multiple  of  72  must  contain  3  twice  as  a  factor. 
Hence,  the  number  24  x  32  x  5  =  720  contains  all  the  fac- 
tors of  the  three  numbers  48,  60,  72.  Therefore  the 
L.  C.  M.  of  48,  60,  72,  is  720. 

To  find  the  L.  C.  M.  of  two  or  more  numbers,  resolve 
each  of  the  numbers  into  its  prime  factors,  then  find  the 
product  of  all  the  prime  factors  of  the  given  numbers, 
taking  each  factor  the  greatest  number  of  times  it  occurs 
in  any  of  the  numbers. 

Another  Method 
Example  l.   Find  the  L.  C.  M.  of  48,  60,  72. 


2 
2 
2 
3 

48 

60 

72 

24 

30 

36 

12 

15 

18 

6 

15 

9 

253 
L.  C.  M.  =3x5x2x3x2x2x2  =  720. 


32  ADVANCED  BOOK  IN  ARITHMETIC 

Step  1.    Arrange  the  numbers  in  a  horizontal  row. 

Step  2.  Divide  by  a  prime  factor  common  to  two  or 
more  of  the  numbers.  Set  down  the  quotients  and  the 
undivided  numbers. 

Step  3.  Treat  the  second  horizontal  row  in  the  same 
manner,  and  so  on  until  a  horizontal  row  is  obtained 
which  contains  numbers  prime  to  one  another.  If,  at  any 
stage  of  the  process,  a  horizontal  row  contains  a  number 
which  is  a  factor  of  some  other  number  in  that  row,  then 
strike  out  such  factor. 

The  continued  product  of  the  numbers  in  the  last  row 
and  of  the  divisors  will  be  the  L.  C.  M. 

EXERCISE   19 
Find  the  L.  C.  M.  of: 

1.  16,  20.  16.  60,  96.  31.  15,  20,  25. 

2.  21,  14.  17.  84,  108.  32.  12,  18,  20. 

3.  18,  60.  is.  55,  77.  33.  45,  63,  70. 

4.  18,  45.  19.  54,  90.  34.  14,  35,  40. 

5.  21,  49.  20.  72,  108.  35.  12,  16,  18. 

6.  28,  70.  21.  75,  125.  36.  14,  24,  40. 

7.  42,  56.  22.  36,  54,  72.  37.  15,  24,  25. 

8.  36,  54.  23.  36,  90,  60.  38.  77,  143,  22. 

9.  34, 51.  24.  48,  64,  36.  39.  18,  20,  45. 

10.  48,  72.  25.  12,  15,  18.  40.  30,  70,  105. 

11.  96,  120.  26.  14,  21,  35.  41.  30,  40,  48. 

12.  28,  30.  27.  30,  35,  21.  42.  21,  28,  35. 

13.  32,  80.  28.  28,  42,  70.  43.  12,  18,  27. 

14.  26,  39.  29.  32,  35,  150.  44.  12,  15,  16,  18. 
is.  48,  84.  30.  30,  45,  48.  45.  36,  40,  45. 


FRACTIONS  33 

FRACTIONS 

If  23  be  divided  by  4,  the  process  is  indicated  %£- ;  the 
quotient  is  5-|.  This  means  4  is  contained  in  23,  5  times 
and  3  remains  to  be  divided  by  4.  From  questions  of  this 
character  the  term  fraction  (Latin  fractus,  broken  in  pieces) 
has  arisen.  -^-,  5f,  f,  are  all  fractions. 

A  Fraction  is  an  indicated  Division. 

For  purposes  of  instruction  fractions  are  regarded  from 
another  point  of  view. 

If  the  rectangle  ABOD  is  divided  into  four  equal  parts 
by  lines  having  the  same  direction  as  AB,  one  of  these 
parts  is  called  one  fourth  of  the  whole  rectangle ;  two 
of  the  parts  are  called  two  fourths  of  the  rectangle ; 
three  of  the  parts  are  called  three  fourths  of  the  rec- 
tangle ;  and  four  of  the  parts  are  called  four  fourths 
of  the  rectangle.  In  general,  if  any  one  thing  is  divided 
into  four  equal  parts,  one  of  the  parts  is  called  a  fourth ; 
two  of  the  parts  are  called  two  fourths ; 
three  of  the  parts,  three  fourths,  etc. 
Similarly,  if  any  thing  is  divided  into 
five  equal  parts,  one  of  the  parts  is 
called  one  fifth ;  two  of  the  parts  are 
called  two  fifths ;  three  of  the  parts, 
three  fifths,  etc. 

In  the  above  rectangle,  if  the  line  EF  is  drawn  so  as  to 
divide  AB  and  CD  each  into  two  equal  parts,  the  whole 
figure  will  be  broken  up  into  eight  rectangles;  one  of 
these  rectangles  is  one  eighth  of  the  whole ;  two  of  them 
are  two  eighths  ;  three  of  them,  three  eighths,  etc.  Divide 
AE  and  also  EB  into  three  equal  parts  and  draw  through 
the  points  of  division  lines  parallel  to  BO.  What  part  of 
ABOD  is  one  of  the  small  rectangles  ?  two  of  them  ?  etc. 


34  ADVANCED  BOOK  OF  ARITHMETIC 

NOTATION  OF  FRACTIONS 

1  eighth  is  written  -|.  2  eighths  is  written  f . 

3  eighths  is  written  f .  4  eighths  is  written  |. 

5  eighths  is  written  f .  6  eighths  is  written  |,  etc. 

How  many  thirds  are  in  1  thing  ? 
How  many  fourths  are  in  1  thing  ? 
How  many  sevenths  are  in  1  thing  ? 
How  many  eighths  are  in  1  thing  ? 
How  many  tenths  are  in  1  thing  ? 

In  the  notation  of  fractions,  what  does  the  number  be- 
low the  line  indicate  ? 

What  does  the  number  above  the  line  indicate  ? 

If  a  unit  quantity  is  divided  into  any  number  of  equal 
parts,  one  of  these  parts  is  called  a  fractional  unit,  or  unit 
fraction. 

A  fraction  is  one  fractional  unit,  or  two  or  more  fractional 
units  of  the  same  denomination. 

A  fraction  is  expressed  by  two  numbers,  one  number 
being  written  above  a  horizontal  line  and  the  other  num- 
ber being  written  below  the  same  horizontal  line. 

The  number  above  the  horizontal  line  is  called  the 
numerator,  because  it  numbers  the  parts  taken,  i.e.  tells 
how  many  fractional  units  there  are. 

The  number  below  the  line  is  called  the  denominator : 
it  names  the  fractional  unit,  and  indicates  how  many  frac- 
tional units  there  are  in  the  unit  quantity  from  which  the 
fractional  unit  is  derived. 

Thus,  |  signifies  the  unit  quantity  is  broken  into  4 
equal  parts  and  3  of  these  parts  are  taken.  Here  the 
fractional  unit  is  \  (one  fourth) ;  3  is  the  numerator,  and 
4  is  the  denominator. 


FRACTIONS  35 

The  numerator  and  denominator  are  called  the  terms  of 
the  fraction. 

A  proper  fraction  is  one  whose  numerator  is  less  than 
its  denominator.  Thus,  £  is  a  proper  fraction  because  4 
is  less  than  7. 

An  improper  fraction  is  one  whose  numerator  is  greater 
than  or  equal  to  its  denominator.  Examples  :  -^-,  -£. 

A  mixed  number  is  a  number,  part  integral  and  part 
fractional.  Thus,  4|  is  a  mixed  number. 

Read  and  explain  what  each  represents : 

*•  i  f  <  I'  f '  I'  I'  f'  I  -¥->  ¥>  fr  *.  l<  f '  ¥-'  f  <  J.  *.  V-  f 
f ,  f  v-,  ¥'  |,  l,  |,  |,  V-.  -¥-'  iV  A-  H'  ft- 

Is  ^  of  1  week  equal  to  ^  of  2  weeks  ? 

Is  |^  of  1  week  equal  to  ^  of  3  weeks? 

Is  ^  of  1  week  equal  to  ^  of  4  weeks? 

Isf  of  $1  equal  to  i  of  $2? 

Is  |  of  $1  equal  to  £  of  f  3? 

Is  |  of  $1  equal  to  1  of  $4? 

Is  |  of  1  foot  equal  to  \  of  3  feet  ? 

The  above  illustrations  show  that  a  fraction  may  be 
read  in  two  ways.  For  example,  f  may  be  read  two  fifths, 
or  two  divided  by  five ;  ^  may  be  read  four  sevenths,  or 
four  divided  by  seven. 

REDUCTION  OF  FRACTIONS 

Is  If  feet  =  |  of  1  foot  ?     Is  1|  =  |  ? 
Is  f  of  1  hour  =  41  hours  ?     Is  |  =  4£  ? 
Is  -|-  of  an  apple  =  |  of  an  apple  ?     Is  ^  =  f  ? 
Changing  the  form  of  fractions  without  changing  their 
values  is  called  reduction  of  fractions. 


36  ADVANCED  BOOK  OP  ARITHMETIC 

EXERCISE  20 
Reduce  to  integers  or  mixed  numbers  : 

1.  Jf.  10.    -%°-.  19.    -2f .  28.  ff. 

2.  -\3-.  11.    -£g9-.  20.    f|.  29.  -££. 

3.  \t.  12.    1|.  21.    ||.  30.  \°-. 

4.  ^.  13.    If  22.    ff.  31.  If. 

5.  ^g1-.  14.    -4^.  23.     ff.  32.  ||. 

6.  \2-.  15.    1£°.  24.    If.  33.  -Ug8-- 

7.  ^-.  16.     -1^°-.  25.     ff.  34.  If-. 

8.  ^°-.  17.    \22°-'  26'     if  35«  H^' 

9.  Y--  18'    T|-  27>     '/•  36-  -W- 

Reduce  to  an  improper  fraction  4|. 
1    =    6  sixths. 
4    =  24  sixths. 
4|  =  29  sixths  =  2/-. 

Another  Method 

What  number  divided  by  6  gives  4  as  a  quotient  and 
5  as  a  remainder  ? 

The  answer  is  6  times  4  -f  the  remainder,  5.     There- 
fore, 4 1  =  %£-. 

EXERCISE  21 

Reduce  to  improper  fractions : 

1.  3|.  7.  8|.  13.  19f  19.  201f  25. 

2.  3|.  8.  5|.  14.  18|.  20.  16^f.  26. 

3.  5f.  9.  9T\.  15.  27T\.  21.  14|.  27.     14|. 

4.  9f.  10.  IQlf  16.  38-^.  22.  15f.  28.     37f. 

5.  9f.  11.  11TL.  17.  39f  23.  19|.  29.    44T\. 

6.  10T\.  12.  10^.  18.  19f.  24.  291.  30. 


FRACTIONS 


37 


How  many  squares  are  in  the  rectangle  ABCD? 
How  many  are  in  |  of  it  ?  in  -J-  of  it  ? 
In  |  of  it?  in  \  of  it?  in  f  of  it? 
In  1  of  it?  in  \  of  it?  in  \  of  it? 

555 

In  |  of  it?  in  \  of  it?  in  f  of  it? 

In  TL  of  it?  in  ^  of  it?  in  -^  of  it? 

In  T9o  of  it?  in  ^  of  it?  in  ^  of  it? 

How  many  squares  are  in  -$-§  of  the  rectangle  ABCD? 

How  many  squares  are  in  -|  of  the  rectangle  AB  CD  ? 

How  do  the  fractions  f  and  -f$  compare  ? 

How  may  the  fraction  -f$  be  obtained  from  ^  ? 

How  many  squares  are  in  %  of  the  rectangle  ABCD  ? 

How  many  squares  are  in  J|-  of  the  rectangle  ABCD? 

How  do  !"!  and  ^  compare  ? 

How  may  ^  be  obtained  from  J-|  ? 

From  the  above  rectangle  can  it  be  shown  that  : 


The  terms  of  a  fraction  may  be  multiplied  or  divided  by  the 
same  number  and  the  value  of  the  fraction  remains  unchanged. 


38  ADVANCED  BOOK  OF  ARITHMETIC 

Example.     Reduce  f-  to  fourteenths. 

f  =  -£%.     The  result  is  obtained  by  multiplying  the  terms 
of  f  by  2. 

EXERCISE  22 
Reduce  : 

1.  |  to  9ths,  to  15ths;  to  24ths;  to  SOths;  to  36ths. 

2.  f  to  8ths;  to  16ths;  to  24ths;  to  32ds;  to  40ths. 

3.  |  to  lOths;  to  20ths;  to  25ths;  to  35ths;  to  40ths. 

4.  |  to  14ths;  to  21sts;  to  28ths;  to  35ths;  to  49ths. 

5.  f  to  16ths;  to  24ths;  to  40ths;  to  48ths  ;  to  64ths. 

6.  f  to  32ds  ;  to  48ths  ;  to  56ths  ;  to  72ds  ;  to  SOths. 

7.  I  to  18ths  ;  to  27ths  ;  to  45ths  ;  to  63ds  ;  to  72ds. 

8.  -^  to  20ths;    to  50ths;    to   70ths;    to    SOths;    to 
90ths. 

9.  ^  to  24ths;  to  48ths;  to  72ds;  to  84ths;  to  96ths. 

10.  3  to  7ths  ;   to  lOths  ;  to  12ths  ;  to  20ths. 

11.  5  to  4ths;  to  9ths;  to  14ths;  to  25ths. 

12.  7  to  3ds  ;   to  8ths  ;   to  lOths  ;   to  12ths. 

Reduce  f  to  28ths.  f  =  f|.  Do  this  by  multiplying 
the  terms  of  the  fraction  ^  by  7.  Conversely,  reduce 
||^  to  |  by  dividing  the  terms  of  the  fraction  |-^  by  7. 

A  fraction  is  said  to  be  in  its  simplest  form  when  its 
terms  are  integers  prime  to  each  other. 

A  fraction,  in  its  simplest  form,  is  also  said  to  be  in  its 
lowest  terms. 


Example.  Reduce  to  lowest  terms 
1^0  =  If  =  yf  =  •£.  Dividing  the  terms  of  the  fraction 
by  2,  the  result  is  |~|.  Dividing  the  terms  of  this  fraction 
by  4,  the  result  is  if.  Dividing  the  terms  of  ^|  by  3,  the 
result  is  -|. 


FRACTIONS  39 


EXERCISE  23 

Reduce  to  lowest  terms  : 

i-  T62<  A>  ii'  If  A'  if  I*.  fi  I  f  • 

2.  if.  iMMMMMt'  t£- 

3.  M,  If,  *f  ,  If,  f  I,  It  If,  i\V 

*.     Afe  T4oV  T63°5>  tt<  At  At  '«|,  ffl* 


ADDITION 

Example  l.    Add  J,  J. 

In  adding  fractions,  select,  as  a  matter  of  convenience, 
the  lowest  denomination  common  to  the  fractions. 

2         6  " 

1—2 
3~6- 

Hence,  i+i  =  |  +  |  =  |. 

Example  2.    Add  |,  |. 

Here  the  lowest  denomination  common  to  both  fractions 
is  12ths.     Notice,  12  is  L.  C.  M.  of  4,  6. 

f  =  T92- 

f-lf 

Hence,  f  +  f  =  &  +  {%  =  ||  =  1  A- 

Example  3.    Add  4^,  2f,  1^. 

First  add  the  fractions  J,  f  ,  y5^.     To   do 
12~     this,  find  the  L.  C.  M.  of  the  denominators. 

:      The  L-  c-  M-  °f  2'  6'  i2'  is  12- 


the  sum  is  f|-  =  |  =  lf. 

Write  |  in  the  sum  and  carry  1.    Add  next  the  integers 
4,  2,  1,  and  the  1  carried.     The  sum  is  8f  . 


40  ADVANCED   BOOK  OF  ARITHMETIC 

To  add  fractions,  first  reduce  them  to  equivalent  fractions 
having  the  same  denominator.  Then  add  the  numerators, 
and  underneath  the  sum  write  the  common  denominator. 

If  the  resulting  fraction  is  not  in  its  simplest  form, 
reduce  it  to  its  simplest  form. 

To  add  mixed  numbers,  first  add  the  fractions,  and  to 
this  sum  add  the  sum  of  the  integers. 

EXERCISE  24 

Find  the  sum  of  : 

1.  1   ,J,1.  15.    J,J,i,ft.  29.    f,^,  f. 

2.  M,f.  I*'  M.J.&-  30.  M.&.H- 

3.  l,i,f.  17.  TVM-  31,  1J,2J,8|. 

4.  1    f,  |.  18.  I|<  f,  f  32.  21    5J,  7&. 

5.  J,  |,  J_.  19.  f,TVlf  33.  31,  21   2f 

6.  $,£,£.  20.  f,T\,M-  34. 

7.  1    ^i-  21.  |,^,H-  35'  9t.7|,10 

8.  J5,  |,  J.  22.  J,  |,  ft.  36.  10&,  9f, 
9-  f  iJ-  23.  i,|,TVH.  37.  91 

10.  f,i,iV  24.  |,^,  If  38. 

11.  iVi'f  25.  J,J,Tk&.  39. 

12.  TV|,1  26.  |,|,|.  40.    41,51,711. 

13.  |,f.  27.  |,1,^.  41. 

14.  5V,^,f.  28.  |,|,TV  42. 

43.  A  boy  has  $  £  and  f  |.     What  part  of  $  1  has  he  ? 

44.  How  much  is  |  of  an  hour  ?  ^  of  an  hour  ?  ^  of  an 
hour? 

45.  Which  is  the  largest  and  which  the  smallest  of  the 
three  fractions,  f »  f >  f  ? 


FRACTIONS  41 


SUBTRACTION 
Example  1.    From  |  take  f . 


_  is 
—  Tl 


tf-tt-* 

Example  2.    From  17^  take  12|. 


Reduce   the  fractions   to   20ths.  |-|  cannot  be  taken 

from  £$.     Take  it  from  12%  ;   that  is,  from  —  ^t  —  .     Carry 

^jU 

I.  1  and  12  are  13  ;  13  and  4  are  17.     The  remainder  is 
4H- 

EXERCISE  25 
Find  the  value  of  : 

1.  f-  i             15.    5-3f.  29.    31Jf-20TV. 

2.  |  -  i.            16.    6  -  4^.  30.    40f  -  30|i. 

3.  f-1                 17.     11  -f  31.     41  -If. 

4.  §  -  |.               18.     13  -  |.  32.     7|  -  3|. 

5.  &-J.            19.     14-51.  33.     91-5TV 

6.  |  -  f  .                20.     15  -  B-^.  34.     101  -  9f  . 

7.  T%-f.          21.   28-21TV  35.   16|-9|. 

8.  11  -|.            22.    17|-101.  36.    41-1^. 

9.  ^-f            23.     ISf-llf  37.    9f-2TV 

10.   y-|.          24.   5|-2TV  38.   81-3-^. 

II.  if-f.          25.    29|-24f.  39.    9f-4li. 

12.  |-TV            26.    33f-171.  40.     18J-9f. 

13.  3  -  11.          27.    9^  -  3f  41.    28J  -  8|. 

14.  4  -If.          28.    32||-30TV  42.    17i-92V 


42  ADVANCED   BOOK  OF  ARITHMETIC 

43.  What  number  must  be  added  to  1|  to  make  7-|? 

44.  A  man  buys  a  suit  of  clothes  for  $12-|  and  gives  3 
five-dollar  bills  in  payment.     How  much  change  should 
he  receive? 

45.  From  a  piece  of  cloth  containing  17f  yards  14| 
yards  are  sold.     How  many  yards  are  left? 

46.  A  boy  buys  two  books  costing  $ f  and  $-|.     How 
much  change  should  he  get  out  of  a  $  2^  gold  piece  ? 

MULTIPLICATION   OF   FACTORS.     CANCELLATION 

Is  2x3x4x9=  (2  x  3)  (4  x  9)  =  (2  x  9)  x  (3  x  4)  ? 

The  product  of  any  number  of  factors,  no  matter  how  the 
factors  are  grouped,  is  the  same.  This  is  the  Associative 
Law. 

Is  5  x  (2  x  3  x  4  x  9)  ==  10  x  3  x  4  x  9>=  2  x  15  x  4  x  9 

=  2x3x20x9  =  2x3x4x45? 

Is  (2x3x4x9) -2  =  2x3x^2x9  =  3x4x9? 

A  continued  product  is  multiplied  by  a  number  if  one  of 
its  factors  is  multiplied  by  the  number. 

A  continued  product  is  divided  by  a  number  if  one  of  its 
factors  is  divided  by  the  number. 

Cancellation  is  the  shortening  of  the  process  of  division 
by  dividing  dividend  and  divisor  by  the  same  factor  or 
factors. 

Find  by  cancellation  the  quotient : 

18x27xl6  =  2x9x2 
9x8x3        Ixlxl* 

Dividing   9   into    18,    8    into    16,    and    3    into    27,   the 

quotient  is  2  x9  X  2  =  36. 
Ixlxl 


FRACTIONS  43 

EXERCISE  26 

Find  by  cancellation  the  quotient : 

4  x  18  x  3  x  24  22  x  88  x  15 

9  x  4  x  144  132  x  4 

18  x  90  x  105  1760  x  99 

2.     — — — — — — — •  9. 


14  x  25  x  3  4  x  88  x  165 

27  x  64  x  8  5280  x  14 

o.     —    ^^ — - — •  1O. 


108  x  32  176  x  84 

343  x  125  1728  x  34 


35  x  35  27  x  136  x  8 

16  x  16  x  8  x  81  640  x  5200 


64  x  32  x  3  125  x  512  x  13 

225  x  216  2380  x  104 

o. 


75  x  9  x  12  119  x  8  x  13 

108  x  27  x  121  111  x  39  x  12 

22  x  33  x  18  "  74  x  27  x  13  * 

15.  A  farmer  exchanged  320  acres  of  land  worth  $50 
an  acre  for  25  city  lots.   Find  the  price  of  a  lot. 

(price  of  a  lot)  x  25  =  $50  x  320. 

Hence,  price  of  lot  =  $5Qx32°  =  $640. 

2o 

16.  How  many  cows  at  $40  a  head  cost  as  much  as  15 
horses  at  $64  a  head? 

17.  How  many  dozen  eggs  at  35^  a  dozen  must  be  sold 
to  pay  for  7  barrels  of  apples  at  $2.10  a  barrel  ? 

18.  A  laborer  receives  $3.20  a  day.     How  many  days 
must  he  work  to  pay  for  6  tons  of  coal  at  $8  per  ton  ? 

19.  A  bicyclist  rides  at  the  rate  of  9  miles  an  hour. 
How  long  will  it  take  him  to  travel  as  far  as  a  train  goes 
in  6  hours  at  the  rate  of  33  miles  an  hour  ? 

20.  How  many  cattle  at  $42  a  head  must  be  sold  to  pay 
for  11,200  bushels  of  wheat  at  75  j*  a  bushel  ? 


44  ADVANCED  BOOK  OF  ARITHMETIC 

MULTIPLICATION 

Multiplication  of  a  fraction  by  an  integer. 

Example  1.    Multiply  f  by  18. 

I  x  18  =  3  fourths  x  18  =  54  fourths  =  *£  =  13|  =  131- 

Example  2.    Multiply  3T3^  by  14. 

3T3_  x  14  =  (3  x  14)  +  (^  x  14)  =42  +  4i  =  46f 

To  multiply  a  mixed  number  by  an  integer,  first  multiply 
the  fractional  part  of  the  mixed  number  by  the  multiplier, 
next  multiply  the  integral  part  by  the  multiplier  ;  add 
the  two  results  for  the  final  product. 

EXERCISE  27 

Find  the  value  of  : 

1.  I  x  18.        9.  T3¥  x  24.  17.  If  x  10.  25.  7T\  x  33. 

2.  |  x  14.      10.  T5¥  x  40.  is.  1|  x  12.  26.  8T\  x  19. 

3.  I  x  7.        11,  T9g  x  34.  19.  3f  x  10.  27.  9T52  x  80. 

4.  -&  x  18.    12.  11  x  42.  20.  5^  x  54.  28.  6ii  x  102. 

5.  T^  x  16.    is.  if  x  44.  21.  6|  x  36.  29.  7T7e  x  60. 

6.  f  X  12.        14.    If  X  50.      22.    91  X  16.          30.    11^  X  15. 

7.  |x!9.      is.  i£x45.     23.  7f  x44.        31.  12^x18. 

8.  |  X  28.        16.    If  X  9.         24.    10T*g  X  24.     32.    lOf  X  13. 

33.  Find  the  cost  of  a  dozen  cans  of  baking  powder  at 
37^  cents  a  can. 

34.  A  pail  of  mackerel  cost  21  dollars.     Find  the  cost 
of  20  pails. 

35.  Find  the  cost  of  a  barrel  of  sugar  weighing  325 
pounds,  if  the  cost  per  pound  is  411/. 

36.  When  starch  sells  for  3^  a  pound,  find  the  price 
of  15  pounds  of  starch. 


FRACTIONS  45 

37.  When  wheat  is  79|  cents  a  bushel,  how  much  will 
164  bushels  bring  ? 

38.  Find  the  cost  of  a  75-pound  chest  of  Hyson  tea  at 
42|  ^  per  pound. 

39.  If  a  dozen  cakes  of  yeast  cost  42|^,  find  the  cost  of 
9  dozen  cakes  of  yeast. 

40.  Pepper  sells  for  14|^  a  pound.     Find  the  cost  of  2 
bags,  each  containing  120  pounds. 

41.  A  pound  package   of  chocolate  costs  31J^.     Find 
the  cost  of  25  such  packages. 

42.  A  square  rod  equals  30^  square  yards.     Reduce  160 
square  rods  to  square  yards. 

43.  A  link  of  a  surveyor's  chain  is  7||  inches.     If  the 
chain  contains  100  links,  how  many  inches  long   is   the 
chain  ? 

44.  When  silk  sells  at  $  |-J  a  yard,  what  is  the  cost  of 
14  yards  of  silk  ? 

45.  A  degree  on  a  meridian  of  the  earth's  surface  is 
about  69^  miles.     How  many  miles  are  in  15  degrees  ? 
in  40  degrees  ?  v 

46.  A  person  fails  for  $  9,800.     His  creditors  receive  $| 
on  every  dollar  that  he  owes.     How  much  in  all  do  they 
receive  ? 

47.  A  mass  of  copper  and  lead  weighs  2,240  pounds;    f 
of  the  mass  is  copper.     How  much  copper  and  how  much 
lead  is  in  the  mass  ? 

48.  A  man  invests  $2,300.     At  the  end  of  a   year  his 
gain  is  ^  of  his  investment.     Find  his  gain  and  the  value 
of  the  investment  at  the  end  of  the  year. 


46 


ADVANCED  BOOK  OF  ARITHMETIC 


MULTIPLICATION  OF  A   FRACTION  BY   A   FRACTION 

Multiplication  of  fractions  extends  the  meaning  of  the 
term  "multiplication." 

^  X  f  ,  or  |  of  £,  means  2  times  ^  of  -|. 
•J  X  f  ,  or  |  of  |,  means  3  times  -|  of  J. 
Example  l.    What   is   the   area   of   a   rectangle  whose 
length  is  |-  of  an  inch  and  width  |  of  an  inch  ? 
D  _  E    C      Take  AB  1  inch.     Let  it  be  divided 
into  five  equal  parts.     Construct  upon 
it  a  square  ABCD.     Divide  AD  into 
three  equal  parts.     Draw  lines  through 
the   points   of   division.      KNED   will 
have  for   its   dimensions  |-  of  an  inch 
and  J  of  an   inch.     By  counting  the 
small  rectangles  in  KNED  the  number 
is  found  to  be  eight  (4  x  2),  and  the  number  in  ABCD 
is  fifteen  (5x3).     Hence,  the  area  of  KNED  is  T8^  of  a 
square  inch. 

1  of  f  =  iV     Therefore,  2  times  J  of  |  =  ^.     (Show 
by  figure.) 

PBOCESS.  x     =    £-' 


Example  2.  Find  the  area  of  a  rectangle  3^  by  2T3^  feet. 


To  multiply  a  fraction  by  a  fraction,  take  the  product  of 
the  numerators  for  the  numerator  of  the  product,  and  the 
product  of  the  denominators  for  the  denominator  of  the 
product. 

To  multiply  mixed  numbers,  reduce  them  to  improper 
fractions,  and  then  apply  the  rule  for  multiplication  of 
fractions. 


FRACTIONS  47 

EXERCISE   28 

Multiply  : 

1.  f  f  .  15.    T%,  ||,  _V  29.    71   7J,  ^. 

2.  f  |.  16.    ^,  |f  |J.  30.    3f  3f  &. 
3-    |,  if                   "•    if  T\°r  If-  31.    51   5f  T\. 

*•  f  A-  18-  ii  &  if-        32-  7I'  3^'  &• 


5.  |,  If  19-  1,  i  ^T-  33.  If  £,  |  . 

6.  £,  J|.  20.  |,  f,  Iff.  34.  7|,  If  ||. 

7.  1|,  ff  21.  If,  If.  35.  41,  4J,  3f  . 

8.  H,  Vft.  22.  1|,  1J,  1J.  36.  If  If  f. 

9.  ^,  J&.  23.  2f  ^,  f  .  37.  20J,  20}. 
10.  If,  |f.  24.  3|,  T\,  f  38.  4f  5f  1T3T. 

U-  M'  *•  25-  2f'  2!<  A-  39.  6|,  If  ^. 

12.  if,  ^  |.  26.  7f  1|,  If.  40.  2|,  1TV  If 

13.  f,  if  A.  27.  2f,  3f,  If  41.  2f,  8f, 

14.  f  15,  |f  28.  41,  2|,  |f  42.  6f  4|, 

43.  Find  the  value  of  63|  acres  of  land  at  $55^  an  a,cre. 

44.  Find  the  cost  of  12  J  pounds  of  meat  at  15^  cents 
per  pound. 

45.  Find  the  price  of  4J  bushels  of  wheat  at  81^  cents 
per  bushel. 

46.  A  speculator  buys  10,000  bushels  of  wheat  at  79| 
cents  per  bushel  and  sells  it  when  wheat  is  81^  cents  per 
bushel.     Find  his  profit. 

47.  Coal  cost  $9J  a  ton.     Find  the  price  of  5J  tons. 


48  ADVANCED   BOOK  OF  ARITHMETIC 

DIVISION  AND  RATIO 

In  division  the  product  of  two  numbers  and  one  of  the 
numbers  are  given,  and  the  other  number  is  sought. 

Example.     Divide  1^  by  J. 

f  of  the  quotient  =  1^  =  -J^6-. 
Therefore,  |  of  the  quotient  =  -|-  of  -1^6-. 
Therefore,  f  of  the  quotient  =  |  of  ^6  . 
Therefore,  the  quotient  =  f  of  -1/  =  2f  . 
Observe  that  -^  is  divided  by  f  by  multiplying  by  |. 

Hence  the  rule  for  division:  Invert  the  terms  of  the 
divisor  and  then  proceed  as  in  multiplication. 

If  the  product  of  two  numbers  is  unity,  either  is  called 
the  reciprocal  of  the  other. 

ILLUSTRATIONS 

7  x  7  =  1.     The  reciprocal  of  7  is  ^,  and  of  ^-  is  7. 
|  x  f  =  1.     The  reciprocal  of  4J  is  |,  and  of  |  is  4J. 
185  x  ~V~  =  !•     The  reciprocal  of  T8^  is  -^-,  and  of  -^5-,  or  1|, 


Hence  the  rule  for  division  may  be  briefly  stated  : 
Multiply  the  dividend  by  the  reciprocal  of  the  divisor. 

By  the  ratio  of  one  number  to  another  number  is  meant 
the  quotient  of  the  first  number  by  the  second.  Thus  the 

ratio  of  9  inches  to  12  inches  is    9  mches  =  |  . 

12  inches 

The  ratio  of  9  inches  to  12  inches  is  briefly  indicated 
9  in.:  12  in. 

Example.     Find  the  value  of  the  ratio  4  days  :  7^  hours. 
4  days  =  4  x  24  hours  =  96  hours. 
96  hours  -f-  7-J  hours  =  96  -?-  7£  =12.8. 


FRACTIONS  49 

EXERCISE  29 


Divide: 

1. 

if  by  27. 

25. 

12by|. 

49. 

33  by  ^T. 

2. 

1  by  A- 

26. 

9A  by  45. 

50. 

7T93  by  80. 

3. 

1  by  TV 

27. 

4*  by  9. 

51. 

1  by  7f 

4. 

f  o  by  16. 

28. 

14  by  f 

52. 

40  by  ^« 

5. 

3f  by  10. 

29. 

16|  by  63. 

53. 

5lf  by  50. 

6. 

T3obyTV 

30. 

4f  by  If. 

54. 

1  by  9f  . 

7. 

&  by  12. 

31. 

16  by  f 

55. 

60  by  f  . 

8. 

21  by  3. 

32. 

181  by  26. 

56. 

8|  by  46. 

9. 

A  by  if 

33. 

8f  by  2f-. 

57. 

lOf  by  2f  . 

10. 

if  by  16. 

34. 

18  by  f 

58. 

11  by  |. 

11. 

3f  by  8. 

35. 

17f  by  75. 

59. 

A  by  f  . 

12. 

fbyf 

36. 

4|  by  9|, 

60. 

14|  by  111. 

13. 

it  by  15. 

37. 

21  by  f. 

61. 

Ibyi. 

14. 

4T\  by  14. 

38. 

lOf  by  48. 

62. 

ibyTV 

15. 

T9*byTV 

39. 

H  by  31 

63. 

Mbyi^. 

16. 

ii  by  25. 

40. 

25  by  f  . 

64. 

Ibyf. 

17. 

5  ft  by  21. 

41. 

9f  by  52. 

65. 

T9oby^. 

18. 

A*y&- 

42. 

4TV  by  17f 

66. 

i|bylT\. 

19. 

IA  by  21. 

43. 

26  by  f 

67. 

ibyA- 

20. 

9^  by  46. 

44. 

9^  by  75. 

68. 

ibyf. 

21. 

A  by  if 

45. 

3*byf 

69. 

9JT  by  111 

22. 

M  by  36. 

46. 

26  by  if. 

70. 

lby,V 

23. 

9^by77. 

47. 

81  by  15. 

71. 

fbyf 

24. 

22lf  by  8. 

48. 

1  by  4|. 

72. 

7{  by  41 

50  ADVANCED  BOOK  OF  ARITHMETIC 

EXERCISE  30 

1.  What  is  the  ratio  of  6  inches  to  12  inches? 

2.  What  is  the  ratio  of  1  foot  to  1  yard  ? 

3.  What  is  the  ratio  of  1  square  foot  to  1  square  yard  ? 

4.  There  are  30|  square  yards  in  1  square  rod.     What 
is  the  ratio  of  1  square  yard  to  1  square  rod  ? 

5.  What  is  the  ratio  of  3  weeks  to  10  days  ? 

6.  What  is  the  ratio  of  1  hour  to  1  minute  ? 

7.  What  is  the   ratio   of   4   days   to   15  hours?  of   a 
minute  to  an  hour  ? 

8.  What  is  the  ratio  of  325  to  100  ? 
Find  the  values  of  the  following  ratios: 

9.  2:J.  15.  6:f.  21.  3f:2|.  27.    lOf :  71. 

10.  3:1.  i6.  2l:3f.  22.  9|:31J.  28.    7^:  6J. 

11.  4  :  \.  17.  7-| :  21|.  23.  5J  :  7f .  29.    41 :  7 \. 

12.  4  :  f .  18.  5| :  4|.  24.  4£ :  21.  30.    7| :  7^. 

13.  5  : 1.  19.  9| :  7£.  25.  51 : 11.  31.    81 :  If. 

14.  7:f.  20.  3f:5f.  26.  61:31.  32.    6-/T :  41. 
Example  l.    Find  the  price  of  2,000  pounds  of  wheat  at 

84  /  a  bushel. 

To  solve  this  question  there  are  two  steps  to  take. 

Step  1.  Find  the  number  of  bushels  by  dividing  the 
number  of  pounds  by  the  number  of  pounds  in  one  bushel. 

Step  2.  Multiply  the  price  of  one  bushel  by  the  num- 
ber of  bushels. 

SOLUTION.     Number  of  bushels  = 

60 

14 


Price  of  the  wheat  »  84X  x 


* 


60 


FRACTIONS  51 

Example  2.    Three  fifths  of  a  man's  money  is  $2,437. 
How  much  money  has  he? 

|  of  his  money  =  12-437. 

I  of  his  money  =  ^|^I  Or  £  of  $2437. 


f  of  his  money  =  x  5  or  |  of  $2437. 


3 

Hence,  his  money  =  $4,061.67  (to  nearest  cent). 

This  method  of  solving  a  problem  is  known  as  the 
analytical  method.  It  is  called  also  the  unit  method, 
because  the  value  of  the  unit  of  the  quantity  under  con- 
sideration is  first  sought  and  from  this  the  value  of  any 
number  of  units  is  then  obtained. 

Note  that  the  answer  is  obtained  by  multiplying  $  2437 
by  f  ,  the  reciprocal  of  -|.  In  this  problem  there  are  given 
the  product  of  two  factors  and  one  of  the  factors.  The 
other  factor  is  sought.  The  problem  is  therefore  one  of 
division. 

EXERCISE  31 

1.  Find  the  price  of  78  acres  of  land  if  25  acres  are 
worth  $1,375. 

2.  When  18  pounds  of  sugar  sell  for  $1,  find  the  cost 
of  45  pounds. 

3.  When  7  bushels  of  wheat  sell  for  $5.95,  how  much 
can  a  person  get  for  255  bushels  ? 

4.  If  5  bushels  of  barley  sell  for  12,  how  much  will 
343  bushels  sell  for  ? 

5.  If  6  barrels  of  flour  are  sold  for  $45,  at  this  rate 
how  much  will  84  barrels  sell  for  ? 


52        ADVANCED  BOOK  OF  ARITHMETIC 

6.  Seven  barrels  of  pork  sell  for  $80.50.     Find  the 
cost  of  50  barrels  of  pork. 

7.  Nine  barrels  of  salt  cost  $11.70.     Find  the  cost  of 
19  barrels  of  salt. 

8.  Eleven  bushels  of  oats  are  sold  for  $4.51.     Find 
the  value  of  168  bushels. 

9.  Six  barrels  of  lard  bring  $115.     How  much  will 
46  barrels  bring  ? 

10.  When  7  yards  of  sheeting  cost  50  fa  find  how  much 
must  be  paid  for  98  yards. 

11.  Six  yards  of  cambric  sell  for  75^.     How  much  must 
be  given  for  34  yards  of  cambric  ? 

12.  Four  yards  of  flannel  cost  $1.16.     How  much  will 
29  yards  of  flannel  cost  ? 

13.  Eight  yards  of  gingham  cost  60  ^.     How  much  will 
103  yards  cost  ? 

14.  Nine  yards  of  cotton  fabric  cost  75^.     How  much 
will  69  yards  cost  ? 

15.  Six  yards  of  cotton  cheviot  cost  $1.     How  much 
will  81  yards  cost  ? 

16.  Five-eighths  of  a  man's  money  is  $75.     How  much 
money  has  he  ? 

17.  Three-fourths  of  the  length  of  a  pole  is  81  feet. 
Find  the  length  of  the  pole. 

18.  The  eighth  and  the  twelfth  of  a  number  are  15. 
What  is  the  number  ? 

19.  A  dealer  sold  -|  of  his  coal  and  had  170  tons  left. 
How  many  tons  had  he  at  first  ? 

20.  The  fourth  part  and  the  sixth  part  of  a  number  are 
25.     What  is  the  number  ? 


DECIMALS  53 

DECIMALS 

It  is  well  to  fix  in  mind  the  following  facts: 

Tenths  occupy  the  first  place  to  the  right  of  the  decimal 
point ;  hundredths,  the  second  place  ;  thousandths,  the 
third  place  ;  ten-thousandths,  the  fourth  place  ;  hundred- 
thousandths,  the  fifth  place  ;  millionths,  the  sixth  place. 

Read  22.234.  Twenty-two  and  two  hundred  thirty -four 
thousandths. 

Write  twenty-four  tenths. 

Write  24  as  if  it  were  an  integer.  Tenths  occupy  the 
first  place  to  the  right  of  the  decimal  point.  Hence,  24 
tenths  is  written  2.4. 

Write  2,304  hundredths. 

Write  2,304  as  if  it  were  an  integer.  Beginning  at  the 
right,  point  off  two  places  for  hundredths.  Hence,  2,304 
hundredths  is  written  23.04.  If  ^ff^-  be  reduced  to  a 
mixed  number,  it  becomes  23^^  ;  that  is,  23.04. 

Write  11  hundred-thousandths. 

Write  11  as  if  it  were  an  integer.  Beginning  at 
the  right,  point  off  five  places  for  hundred-thousandths. 
Hence,  11  hundred-thousandths  is  written  .00011.  Ob- 
serve that  places  having  no  digits  are  filled  in  with 
ciphers. 

Write  five  hundred  and  five  thousandths. 

First  write  five  hundred,  and  then  write  five  thousandths. 
Hence,  five  hundred  and  five  thousandths  is  written 
500.005. 

Write  seven  hundred  eight  thousandths. 

Here  the  number  of  units  is  708;  the  denomination  is 
thousandths.  As  thousandths  occupy  the  third  place  to 
the  right  of  the  decimal  point,  hence  708  thousandths  is 
written  .708. 


54  ADVANCED   BOOK  OF  ARITHMETIC 

MULTIPLICATION  AND   DIVISION  BY  POWERS  OF   TEN 

Consider  the  two  numbers, 

(«)  320.12, 
(6)  3,201.2. 

Both  are  expressed  by  the  same  figures  written  in  the 
same  order.  The  number  (5)  can  be  obtained  from  the 
number  (a)  by  moving  each  figure  one  place  to  the  left. 
But  moving  a  digit  one  place  to  the  left  makes  its  value 
ten  times  as  great,  and,  hence,  moving  several  digits  each 
one  place  to  the  left  makes  the  number  they  represent  ten 
times  as  great. 

The  number  (5)  can  also  be  obtained  from  (a)  by  mov- 
ing the  decimal  point  in  (a)  one  place  to  the  right.  Also 
(a)  can  be  obtained  from  (5)  by  moving  the  decimal  point 
in  (5)  one  place  to  the  left. 

To  multiply  a  number  by  10,  move  the  decimal  point  in 
the  number  one  place  to  the  right. 

To  divide  a  number  by  10,  move  the  decimal  point  in  the 
number  one  place  to  the  left. 

Consider  the  numbers, 

(a)  320.12, 
(J)  32,012. 

The  number  (5)  is  obtained  from  (a)  by  moving  each 
digit  in  (a)  two  places  to  the  left.  This  multiplies  each 
digit  by  100. 

(6)  may  also  be  obtained  from  (a)  by  moving  the  deci- 
mal point  in  (a)  two  places  to  the  right;  also  (a)  from 
(6)  by  moving  the  decimal  point  two  places  to  the  left. 

To  multiply  a  number  by  100,  move  the  decimal  point  in 
the  number  two  places  to  the  right. 


DECIMALS  55 

To  divide  by  100,  move  the  decimal  point  in  the  dividend 
two  places  to  the  left. 

Consider  the  numbers, 

(a)  320.12, 
(6)  320,120. 

(b)  is  here  obtained  from  (a)  by  moving  each  digit  in 
(a)  three  places  to  the  left.  It  can  also  be  obtained  from 
(#)  by  moving  the  decimal  point  in  (a)  three  places  to 
the  right. 

To  multiply  a  number  by  1,000,  move  the  decimal  point  in 
the  number  three  places  to  the  right. 

To  divide  a  number  by  1,000,  move  the  decimal  point  in 
the  number  three  places  to  the  left. 

The  rules  for  multiplying  by  10,000,  100,000,  are  left 
for  the  reader  to  determine. 

Example  1.  Multiply  86.4  by  10,000.  Moving  the 
decimal  point  four  places  to  the  right,  the  number  be- 
comes 864,000. 

Example  2.  Divide  12.3  by  100,000.  Moving  the  deci- 
mal point  five  places  to  the  left,  the  number  becomes 
.000123. 

EXERCISE   32 
Multiply  by  10  : 

1.  120,  14.2,  .1431,  .00012,  1.7320,  .01234. 
Multiply  by  100 : 

2.  173,  172.8,  19.23,  .001237,  8,654,  17.1. 
Multiply  by  1,000  : 

3.  1156,  32.5,  7.123,  .93891,  .01275,  .00011. 


56  ADVANCED   BOOK  OF  ARITHMETIC 

Multiply  by  10,000 : 

4.  345,  34.25,  5.1739,  6.001,  .01793,  .12. 

5.  Divide  each  of  the  following  numbers  by  10;    by 
100;  by  1,000;  by  10,000 ;  by  100,000  : 


32,734 

9,285. 

773 

3,745.3 

325. 

298 

928.49 

127 

72,173.5 

325 

12.792 

17 

99,999.9 

18. 

326 

3,728.3 

7. 

294 

12.7564 

670 

1,201 

1,000 

3,450 

7,100 

Find  the  values  of  the 

following  ratios  : 

6. 

22.3  :  .223. 

22. 

.001: 

10. 

7. 

3.74  :  .374. 

23. 

.005  : 

100. 

8. 

173.2  :  1.732. 

24. 

9.265 

:  926.5. 

9. 

7.3  :  .073. 

25. 

12.325:  1,232.5. 

10. 

1.25  :  .0125. 

26. 

1.534 

:  153.4. 

11. 

9.28  :  .00928. 

27. 

1,001 

:  .1001. 

12. 

11.34  :  .01134. 

28. 

54  :  .054. 

13. 

7.04  :  .0704. 

29. 

792  : 

.0792. 

14. 

100  :  .01. 

30. 

113: 

.0113. 

15. 

1,000  :  .001. 

31. 

79.28 

:  .7928. 

16. 

.012  :  .12. 

32. 

6.45  : 

6,450. 

17. 

1.24  :  124. 

33. 

99.29 

:  99,290. 

18. 

9.53  :  9,530. 

34. 

7.35  : 

73,500. 

19. 

7.1  :  7,100. 

35. 

9.24  : 

92,400. 

20. 

6.5  :  65,000. 

36. 

8.123 

:  .008123. 

21. 

11.79  :  11,790. 

37. 

.04567  :  45.670. 

DECIMALS  57 

ADDITION" 

Find  the  sum  of  3.4,  2.38,  5.005,  6.2374,  11.1. 

3.4  Write  the  numbers  so  that  units  of  the  same 
2.38         denomination  stand  in  the  same  vertical  column. 

5.005  Then  add  as  integers  are  added. 

6.2374     Write  the  decimal  point  in  the  sum  in  the  same 
11.1  vertical   line   with   the   decimal   points   in   the 

28.1224     addends. 

EXERCISE   33 

Add: 

1.  2.2,  .025,  37.3,  5.284,  6.294,  538.1,  77.77. 

2.  3.5,  7.12,  .339,  47.35,  39.28,  .123,  54.275. 

3.  9.28,  11.18,  .999,  39.28,  7.451,  94.354,  98.76. 

4.  12.49,  1.492,  38.75,  53.41,  98.69,  845.5,  892.9. 

5.  .009,  5.976,  40.99,  6.385,  9.278,  8.239,  64.271. 

6.  .098,  9.853,  19.47,  17.392,  28.394,  8.01,  77.47. 

7.  .285,  11.95,  29.99,  94.931,  1.732,  64.6,  78.75. 

8.  11.4,  17.5,  99.37,  15.273,  9.394,  71.3,  92.95. 

9.  1.21,  12.1,  .121,  8.295,  7.777,  68.7,  78.28. 

10.  15.9,  9.158,  91.58,  9.158,  2.293,  84.5,  .139. 

11.  98.5,  11.667,  66.66,  8.394,  9.928,  76.8,  9.359. 

12.  77.8,  88.88,  99.99,  6.325,  7.384,  94.9,  1.798. 

SUBTRACTION 

Find  the  difference  between  4,001  and  1.7003. 

Arrange  the  numbers  so  that  units  of  the 

4001.0000     same  denomination  stand  in  the  same  vertical 

1.7003     column.     Ciphers  may  be  inserted  after  the 

3999.2997     decimal  point  in  the  minuend.     Proceed  next 

as  in  the  subtraction  of  integers. 


58  ADVANCED  BOOK  OF  ARITHMETIC 

EXERCISE   34 

Find  the  remainder  and  verify  your  answer  in  each 
case: 

1.  7.73-6.78.  14.  10.1-7.325. 

2.  9.29-3.47.  15.  9.24-5.3481. 

3.  6.34-1.95.  16.  8.73-4.4444. 

4.  9.82-7.78.  17.  12.32-5.6741. 

5.  7.45-3.59.  18.  19.33-6.2734. 

6.  10.71-7.79.  19.  9.271-4.3847. 

7.  8.94-3.95.  20.  3.213 -.9875. 

8.  5.012-2.9.  21.  4.321 -.73201. 

9.  10.943-7.97.  22.  5.204-1.3256. 

10.  8.325-4.378.  23.  8.731-5.4557. 

11.  8.924-5.938.  24.  9.21-7.2349. 

12.  7.312-2.7.  25.  7.29-3.4551. 

13.  9.419  -  5.57.  26.  6.001  -  5.112. 

27.  From   seven  hundred  four  thousandths  take  two 
hundred  five  ten-thousandths. 

28.  From  five  hundred  ten  thousandths  take  five  hun- 
dred ten-thousandths. 

29.  From  two  thousand  take  two  thousandths. 

30.  How  much  does  one  thousandth  exceed  one  hundred- 
thousandth  ? 

31.  Find  the  difference  between  a  hundred  and  a  hun- 
dredth. 

32.  From  39  tenths  take  39  thousandths. 

33.  From  100  hundredths  take  100  ten-thousandths. 

34.  How  much  must  be  added  to  one  and  five-tenths  to 
make  ten  ? 


DECIMALS  59 

35.  By   how   much   does   175   hundredths  exceed  175 
hundred-thousandths?     What   is   the    ratio    of    the    first 
number  to  the  second? 

36.  By  how  much  does  $1  exceed  1  mill  ? 

37.  By  how  much  does  $2  exceed  15  mills? 

MULTIPLICATION 

Example  l.    Multiply  3.23  by  25. 
3.23  =  323  hundredths. 
323  hundredths  x  25  =  8075  hundredths  =  80.75. 

Example  2.    Multiply  3.23  by  .25. 

Since  the  multiplier  is  y^g-  of  25,  the  product  3.23  x 
.25  =1^  of  3.23  x  25.  ^  of  80.75=  .8075. 

The  mechanical  work  of  multiplying  may  be  performed 
as  follows : 

3  23 

Multiply  as  if  both  numbers  were  integers,  and 

'          point  off  in  the  product,  commencing  at  the  right, 
as  many  places  as  there  are  decimal  places  in  both 


.8075 


multiplicand  and  multiplier. 


Example  3.    Multiply  .32  by  .018. 
Point  off  five  places. 


Another  ^Explanation 
x  TUo  =  nfiflW  =  -00576- 


To  square  a  number  means  to  multiply  the  number  by 
itself  or  to  take  the  number  twice  as  a  factor. 

To  cube  a  number  means  to  take  the  number  three  times 
as  a  factor. 


60  ADVANCED  .BOOK  OF  ARITHMETIC 

EXERCISE   35 

1.  Find  .04  of  $108;  .05  of  $274;  .06  of  $720;  .07  of 
1144. 

2.  Find  .09  of  $34.50;   .3  of  $75.30;  .08  of  $75.80; 
.07  of  $84.70. 

3.  Find  .4  of  $29.75;  .5  of  $69.48;  .6  of  $68.32;  .1 
of  $328.50. 

4.  Find  .125  of  $80.80;   .75  of  $54;   .6  of  $300.50; 
.25  of  $98.84. 

5.  Find   .625    of    $688;    .875   of    $792.80;    .375   of 
$900.80. 

6.  Find  .375  of  84  acres;  .0625  of  64  acres;  .3125  of 
96  acres. 

7.  Find  .1  of  .1;  .3  of  .4;  .3  of  .3;  .01  of  .2  ;  .01  of  1.2. 

8.  Multiply  27.9  by  18.  23.    1.18  x  .1695  =  ? 

9.  Multiply  1,327  by  1.6.         24.    .97  x  .97  =  ? 

10.  Multiply  3,927  by  .46.  25.  .68  x  .68  =  ? 

11.  Multiply  120.01  by  3.6.  26.  .373  x  .373  =  ? 

12.  Multiply  25  by  .017.  27.  .901  x  .901  =  ? 

13.  Multiply  37.5  by  .07.  28.  .803  x  .803  =  ? 

14.  Multiply  11.9  by  2.4.  29.  .693  x  .693  =  ? 
is.  Multiply  182.54  by  1.49.  30.  .1  x  .1  x  .1  =  ? 

16.  Multiply  .286  by  1.96.         31.    .3  x  .3  x  .3  =  ? 

17.  Multiply  92.24  by  2.7.         32.    .4  x  .4  x  .4  =  ? 
is.    .148x1.15  =  ?  33.    .7x.7x.7  =  ? 

19.  .82x.51  =  ?  34.  1.04x1.04x1.04  =  ? 

20.  1.875  x. 32=?  35.  1.06x1.06x1.06  =  ? 

21.  1.78x1.89  =  ?  36.  1.08x1.08x1.08  =  ? 

22.  18.24  x. 95  =  ?  37.  .25x.25x  .25  =  ? 


DECIMALS  61 

38.  .7645  of  the  asphalt  found  in  West  Virginia  is  com- 
posed of  carbon,  .0783  is  hydrogen,  .1346  is  oxygen,  and 
the  remainder  is  ash.     How  much  of  each  constituent  is 
in  254  tons  of  asphalt  ?     Check  your  answers. 

39.  .7217  of  the  asphalt  found  in  Oregon  is  composed 
of  carbon,   .079  of  hydrogen,  .1461  of  oxygen,  and  the 
remainder  of  ash.     Find  the  amount  of  each  in  385  tons 
of  asphalt.     Check  your  answer. 

40.  Multiply  the  square  of  14  by  .7854. 

41.  The  area  of  the  surface  of  a  sphere  is  obtained  by 
multiplying  the  square  of  the  diameter  by  3.1416.     Find 
the  area  of  the  surface  of  the  earth,  taking  the  diameter 
to  be  7,920  miles.     Compare  your  answer  with  the  area 
given  in  your  geography. 

42.  The  moon  is  nearly  2,200  miles  in  diameter.     Find 
the  area  of  its  surface  in  square  miles. 

43.  The  velocity  of  the  earth  in  its  orbit  is  18.5  miles 
per   second.     How  far   does   it   go   in    1    minute?   in  1 
hour? 

44.  A  hurricane  moves  at  the  rate  of  146.6  feet  per 
second.     How  far  does  it  travel  in  1  minute  ?  in  1  hour  ? 

45.  One  meter  =  39.37  inches.      Find  in  inches  the  dif- 
ference between  64  meters  and  70  yards. 

DIVISION 

Before  undertaking  Division,  it  may  be  well  to  lay 
stress  on  the  fact  that  numbers  in  the  decimal  system  of 
notation  may  be  read  in  many  ways.  Thus,  32.25  may 
be  read,  (a)  32  and  25  hundredths ;  (5)  3,225  hundredths  ; 
O)  32,250  thousandths;  (d)  322,500  ten-thousandths; 
(e)  322.5  tenths;  (/)  3.225  tens. 


62  ADVANCED  BOOK  OF  ARITHMETIC 

Example  l.    Divide  1.293  by  8. 

8")1  293000         ^  *nt°  ^  tenths  gives  1  tenth,  with  a  re- 
161625     mainder  4  tenths.    4  tenths  =  40  hundredths ; 
40  hundredths  and  9  hundredths  =  49  hun- 
dredths.    8  into  49  hundredths  gives  6  hundredths,  with 
a  remainder  1  hundredth.     Change  1  hundredth  to  thou- 
sandths, and  proceed  as  before. 
Example  2.    Divide  .01234  by  4. 

4). 012340         The  work  calls  for  no  explanation. 
.003085 

EXERCISE  36 
Divide: 

1.  73.21  by  8.  9.  8.218  by  7.  17.  5.472  by  6. 

2.  3.45  by  4.  10.  3.942  by  6.  is.  8.2548  by  9. 

3.  19.362  by  6.  ii.  6.475  by  7.  19.  .34794  by  9. 

4.  1.791  by  9.  12.  9.143  by  8.  20.  .67356  by  9. 

5.  4.564  by  5.  13.  .1234  by  5.  21.  .999999  by  7. 

6.  3.927  by  8.  14.  .73206  by  6.  22.  7.3745  by  7. 

7.  .015  by  5.  15.  1.1466  by  7.  23.  6.2676  by  6. 

8.  8.846  by  6.  16.  6.2751  by  8.  24.  1.7346  by  7. 
Find   the   difference  between    .07858    and    .078;    also 

find  the  difference  between  .07858  and  .079. 

Hence  .07858  is  nearer  to  .079  than 
.07858     .07900     it   is   to    .078.     If,  therefore,  one  is 
.078         .07858     asked   to   give   the   value   of   .07858 
.00058     .00042     correct  to  three  figures,  write  for  an- 
swer .079. 

Express  .73948  correct  to  three  figures.     Ans.  .739. 
Express  .25764  correct  to  three  figures.     Ans.   .258. 
Whenever  asked  to  give  a  decimal  correct  to  any  num- 
ber of  figures,  discard  the  remaining  figures  if  the  first 


DECIMALS  63 

one  of  them  is  less  than  5;  if  it  is  5  or  more  than  5,  in- 
crease the  last  figure  by  1. 

Example  l.    Divide  .0732  by  .8. 

Make  the  divisor  an  integer  by  moving  the  decimal 
point  one  place  to  the  right.  Make  a  corresponding 
change  in  the  dividend.  This  change  is  equivalent  to 
multiplying  divisor  and  dividend  by  10. 

8). 7320 
.0915 

Example  2.    Divide  12  by  .125. 

Move  the  decimal  point  in  the  divisor  and  in  the 
dividend  three  places  to  the  right,  i.e.  multiply  each  by 

i'000'  96 

125)12000 
1125 
750 
750 

Example  3.    Divide  3.274  by  6.25. 

.523+ 

625)327.400 
3125 
1490 
1250 


2400 

1875 
525 

"Whenever  the  divisor  is  a  decimal,  make  it  an  integer  by 
moving  the  decimal  point  to  the  right.  Make  a  corresponding 
change  in  the  dividend.  After  doing  this,  proceed  in  exactly 
the  same  manner  as  in  long  division  of  integers.  Write  the 
decimal  point  in  the  quotient  in  the  same  vertical  line  -with 
the  decimal  point  in  the  dividend  transformed. 


64  ADVANCED   BOOK  OF  ARITHMETIC 

EXERCISE  37 

Divide: 

1.  2.34  by  .8.  26.  5  by  .004. 

2.  .012  by  .5.  27.  .1  by  .0001. 

3.  3.475  by  .4.  28.  .04  by  .0008. 

4.  1.2348  by  .6.  29.  .32  by  .00128. 

5.  .1798  by  .5.  30.  .45  by  .0018. 

6.  3.144  by  1.2.  31.  .078  by  .00312. 

7.  5.96  by  1.6.  32.  .067  by  .0268. 

8.  3.2903  by  1.3.  33.  .01  by  .8. 

9.  .27  by  .2.  ,        34.  .002  by  1.6. 

10.  5.376  by  1.6.  35.  .018  by  45. 

11.  9.4851  by  1.5.  36.  .54  by  81. 

12.  3.2  by  6.4.  37.  .243  by  1.944. 

13.  20  by  .5.  38.  .216  by  1.44. 

14.  10  by  .16.  39.  5.12  by  .16. 
is.  40  by  .32.  40.  7.29  by  270. 

16.  56  by  1.12.  41.  34.7231  by  .713. 

17.  84  by  5.6.  42.  31.8791  by  3.97. 
is.  392  by  7.84.  43.  .267584  by  2.96. 

19.  100  by  .625.  44.  .348336  by  .492. 

20.  100  by  .008.  45.  .190256  by  .188. 

21.  400  by  .05.  46.  59.4204  by  5,860. 

22.  144  by  .288.  47.  55.9911  by  108.3. 

23.  15.4  by  .616.  48.  .575484  by  54.6. 

24.  .096  by  .192.  49.  .461071  by  122.3. 

25.  1  by  .001.  50.  4.50775  by  123.5, 


DECIMALS  65 

EXERCISE   38 

The  mileage  and  valuation  by  counties  in  Texas  of  the 
St.  Louis  and  San  Francisco  Railway  as  given  by  the  Texas 
Railroad  Commission  for  the  year  1906  are  as  follows: 

COUNTY  MILEAGE  VALUATION 

1.  Collin  19.51  $346,538.13 

2.  Dallas  2.7  53,300.16 

3.  Denton  9.99  188,311.64 

4.  Grayson  27.44  843,427.59 

5.  Hardeman  8.68  183,997.77 

6.  Tarrant  4.56  191,208.29 

7.  Wilbarger  12.77  192,843.01 

Find  the  valuation  per  mile  in  each  of  the  above  counties. 

8.  On  July  16,  1907,  a  contract  for  paving  Broadway, 
Denver,  Colorado,  was  awarded  on  the  following  itemized 
specifications  and  prices: 

3,050  ft.  6"  x  18"  stone  curb  @  $  1.05* 
2,750  yd.  brick  gutter  @  $  2.25 
22,900  yd.  street  asphalt  pave- 
ment @  $  2.25 
704  ft.  oak  header  @  $  .50 
945  ft.  27"  pipe  sewer  @  $  2.40 
580  ft.  24"  pipe  sewer  @  $  2.00 
580  ft.  21"  pipe  sewer  @  $  1.75 
580  ft.  15"  pipe  sewer  @  $  1.10 
398  ft.  12"  pipe  sewer  @  $  .86 
516  ft.  10"  pipe  sewer  @  $  .75 
12  manholes  @  $45.00 
17  catch  basins  @  165.00 
10  M  ft.  lumber  @  $30.00 

Find  the  total  cost. 

* 6"  x  18''  means  6  inches  by  18  inches. 


66  ADVANCED  BOOK  OF  ARITHMETIC 

REDUCTION  OF  FRACTIONS  TO  DECIMALS  AND  REDUC- 
TION OF  DECIMALS  TO  FRACTIONS 

Example  l.    Reduce  -|  to  a  decimal. 

8)7.000 
.875 

Example  2.    Reduce  -fa  to  a  decimal. 
11)7.00000 
.63636+ 

Example  3.    Reduce  -£$fa  to  a  decimal. 

=.016125. 


Divide  numerator  and  denominator  by  1,000  by  moving 
the  decimal  point  three  places  to  the  left;  then  divide  the 
numerator  by  8. 

A  fraction  in  its  lowest  terms  having  for  denominator  a 
number  whose  prime  factors  are  2's  or  5's  or  both  can  always 
be  exactly  expressed  as  a  decimal. 

A  fraction  in  its  lowest  terms  having  for  denominator  a 
number  containing  prime  factors  other  than  2's  and  5's  will 
give  rise  to  a  decimal  which  never  terminates. 

EXERCISE  39 

Reduce  to  decimals: 

1       3     5     JL    _9_    11     13     15       3_ 
*•      8'  ¥'   16'   16'   16'   16'   16'   16* 

2-  A'  A'  ii-  it'  if'  lV  ii- 

3-  T30>   iVk  T^OU'  TITOO 

Q         11        9Q       1Q       Q*7       Q1        ^Q 

4'     ¥0'  H'  ft'  2-0'  f  0'   60'   It' 


e-  f  f  iV'  ^  ft'  A'  ii'  if- 

7<      t'  it'  ill'    94090' 


DECIMALS  67 

Example  l.  Reduce  .0625  to  a  common  fraction.  .0625 
is  read  625  ten-thousandths  ;  ^ff  f  ^  is  read  in  the  same 
way. 

•0625  =  - 


EXERCISE  40 

Reduce  to  common  fractions: 

1.  .3,  .8,  .25,  .125,  .1875. 

2.  .07,  .0125,  .00875,  .0625,  .0075. 

3.  .009,  .0225,  .1125,  .0275. 

4.  .072,  .0104,  .035,  .0119,  .0375. 

5.  .144,  .0504,  .0768,  .162,  .0112. 

6.  .288,  .0176,  .0325,  .0175,  .425. 

7.  .2875,  .3375,  .5125,  .7375. 

EXERCISE  41 

1.  A  man  walks  3  miles  an  hour.     At  this  rate,  how 
long  will  it  take  him  to  walk  12  miles  ? 

2.  A  train  goes  25  miles  an  hour.     How  long  will  it 
take  it  to  go  300  miles  at  this  rate  ? 

3.  A  bicyclist  travels  at  the  rate  of  9  miles  an  hour. 
How  long  will  it  take  him  to  go  60  miles  ? 

4.  How  would  you  find  the  time  to  go  any  given  dis- 
tance, if  you  knew  the  distance  gone  in  a  unit  of  time  ? 

5.  A  man  walks  3.5  miles  an  hour.     At  this  rate,  how 
long  would  it  take  him  to  go  49  miles  ? 

6.  The   distance   from   London   to   Glasgow   is   401.5 
miles.     An  express  train  goes  this  distance  in  8  hours. 
Find  its  rate  per  hour. 

7.  From   London  to  Edinburgh  is  393.5  miles.     The 
daily  mail   train  takes    7.75  hours  to  go  this  distance. 
Find  its  rate  per  hour. 


I   7 


68        ADVANCED  BOOK  OF  ARITHMETIC 

8.  The  Empire  State  Express  goes  from  New  York 
City  to  Buffalo,  a  distance  of  440  miles,  in  8.25  hours. 
Find  its  rate  per  hour. 

9.  The  mail  train  from  Paris  to  Bayonne  goes  486.25 
miles  in  8.983  hours.     Find  its  rate  per  hour. 

10.  The  distance  from  New  York  City  to  Cleveland  is 
568  miles.     A  train  goes   this   distance   in  19.5  hours. 
Find  its  average  speed. 

11.  A  steamer  goes  from  New  York  City  to  Bremen,  a 
distance  of  4235  miles,  in  7.75  days.     Find  its  rate  per 
day.     Also  its  rate  per  hour. 

12.  The  earth  moves  in  its  orbit  at  the  rate  of  1110 
miles  a  minute.     How  many  times  faster  does  the  earth 
move  than  a  train  which  goes  54  miles  an  hour  ? 

13.  A  city  lot  is  worth  $1800.     If  this  sum  is  .75  of 
the  value  of  the  house  on  it,  what  is  the  value  of  the  house  ? 

14.  If  .7  of  a  sum  of  money  is  $  196,  what  is  the  sum  of 
money  ? 

15.  Cast  iron  is  7.2  times  as  heavy  as  water.      How 
many  cubic  feet  of  cast  iron  weigh  as  much  as  6120  cubic 
feet  of  water  ? 

16.  Coal  is  1.3  times  as  heavy  as  water.     How  many 
cubic  feet  of  coal  weigh  as  much  as  546  cubic  feet  of  water? 

17.  There  are  231  cubic  inches  in  a  gallon.     How  many 
gallons  are  in  1  cubic  foot  ?     (1  cu.  ft.  =  1728  cu.  in.) 

18.  If  2000  pounds  of  coal  cost  18.75,  find  the  price  of 
8750  pounds  of  this  kind  of  coal. 

19.  If  3.5  yards  of  cloth  cost  $12.25,  find  the  price  of 
7.5  yards  of  this  cloth. 

20.  If  1.6  yards  of  velvet  cost  $2.88,  find  the  price  of 
9.75  yards  of  velvet. 


FRACTIONS  69 

EXERCISE    42 

1.  What  fraction  of  a  yard  is  1  foot?     What  fraction 
of  a  yard  is  2  feet  ? 

2.  What  fraction   of   1  foot   is   1    inch?    3    inches? 
4   inches  ?  5  inches  ?  7  inches  ?  8   inches  ?  9  inches  ?  10 
inches  ? 

3.  What  fraction  of  1  yard  is  1  inch  ?     What  fraction 
of  a  yard  is  2  inches  ?  3  inches  ?  4  inches  ?  5  inches  ?  6 
inches  ?  9  inches  ?  12  inches  ?  16  inches  ?  17  inches  ?  19 
inches  ?  24  inches  ?  27  inches  ? 

4.  There  are  8  quarts  in  1  peck.     What  fraction  of 
a  peck  is  1  quart  ?     What  fraction  of  a  peck  is  2  quarts  ? 
3  quarts  ?  4  quarts  ?  5  quarts  ?  6  quarts  ? 

5.  What  fraction  of  a  square  yard  is  2  square  feet  ? 
3  square  feet  ?     4  square  feet  ?     5  square  feet  ?     6  square 
feet  ?     7  square  feet  ? 

6.  What  fraction  of  10  is  2  ?     What  fraction  of  10  is  7  ? 

7.  What  fraction  of  11  is  4  ?     What  fraction  of  13  is  9  ? 

8.  What  fraction  of  100  is  80  ? 

9.  Which  of  the  four  fundamental  rules  enables  us  to 
solve  a  problem  of  this  character  :    What  fraction  of   a 
number  is  some  other  number  ? 

10.  If  4  men  can  do  a  piece  of  work  in  7  days,  how 
long  will  it  take  1  man  to  do  the  same  work  ? 

11.  If  a  team  of  horses  can  plow  a  40-acre  lot  in  16 
days,  how  long  will  it  take  4  teams,  working  together,  to 
plow  the  same  lot  ? 

12.  If  a  man  can  do  a  piece  of  work  in  9  days,  what 
fraction  of  the  work  can  he  do  in  1  day  ?  in  2  days  ?  in 
3  days  ?  in  4  days  ?  in  6  days  ? 


70  ADVANCED  BOOK  OF  ARITHMETIC 


COMPLEX  FRACTIONS 

A  complex  fraction  is  a  fraction  one  or  both  of  whose 
terms  contain  one,  or  more  than  one,  fraction. 

2i     3    1+1-4. 
Ihus,  -£i  jyi  —  22'  are  complex  tractions. 

'       ^  5         "5  T"  3" 


21 
Example  1.    Simplify  ^-|- 


SOLUTION.     2f  +  If  =  f  x  T6T  = 
Or,  multiply  numerator  and  denominator  by  any  number 
which  will  make  the  terms  of  the  fraction  integers. 

2|  __  2|  x  6  _  16  _ 

If      If  x6     11"     ir 


Example  2.    Simplify    '2J  ~    *• 

~~~ 


Step  1.  Simplify  the  numerator. 

Step  2.  Simplify  the  denominator. 

Step  3.  Divide  the  result  of  Step  1  by  the  result  of 
Step  2. 

EXERCISE  43 

,  m.  10.  js.    13.  ?Jtzi 


8  u 

9|         '  40| 


9 
'  -~        ' 


I-  +  11— 1£  4  + |  of  11 

16.   L_t_i.    ,,    ^1^.    Z8.    - 

19.  HiH+l2A.  20.  i^l±}l.xljx: 


AREAS  OF  RECTANGULAR  FIGURES 


71 


AREAS  OF  RECTANGULAR  FIGURES 
EXERCISE   44 

1.  The  dimensions  of  a  room  are  40  feet  by  30  feet, 
and  18  feet  high.     How  many  square  yards  are  in  its  walls 
and  ceiling  ? 

2.  Find   the  area,  in  square   yards,  of   the  walls  and 
ceiling  of  a  room  24  feet  by  16  feet,  and  12  feet  high. 

3.  ABQD  is  a  rectangular  plot  of  ground  400  feet  by 
160  feet.     Surrounding  it  is  a  road  15  feet  wide.     Find 
the  area  of  the  road. 


4.  A  rectangular  park,  600  feet  long  by  560  feet  wide, 
has  a  road  surrounding  it.     Find  the  area  of  the  road  if 
its  width  is  24  feet.     Suppose  the  road  is  fenced  in,  how 
many  feet  of  wire  will  it  take  to  go  once  round  ? 

5.  A  rectangular  grass  plot  252  feet  by  180  feet  has  a 
walk  around  it.     The  width  of  the  walk  is  9  feet.     How 
many  flags,  9  inches  square,  will  be  required  to  flag  the 
walk? 

6.  Find  the  area  of  each  of  the  following  rectangles,  in 
square  feet,  correct  to  two  decimal  figures: 

(a)  136  feet  8  inches  by  115  feet  4  inches. 
(J)  225  feet  by  93  feet  10  inches, 
(c)  78  feet  5  inches  by  56  feet  6  inches. 
25  feet  9  inches  by  50  feet  2  inches. 


72  ADVANCED   BOOK   OF  ARITHMETIC 

(e)  104  feet  2  inches  by  153  feet  11  inches. 
(/)  203  feet  by  53  feet  9  inches. 
(#)  223  feet  10  inches  by  78  feet. 
(A)  618  feet  1  inch  by  130  feet  7  inches. 
HINT.     Reduce  the  inches  in  each  example  to  the  fraction  of  1  foot. 

7.  Find  the  area  of  the  following  rectangles,  giving 
the  results  in  square  yards,  correct  to  two  decimal  figures : 

(0)  84. 5  feet  by  76.75  feet. 
(6)  90.67  feet  by  84.33  feet. 
0)  96. 34  feet  by  85. 28  feet. 
(d)  177.33  feet  by  82.54  feet. 
0)  129.55  feet  by  79.63  feet. 

8.  A  cornfield  is  213^  rods  long  and  96  rods  wide. 
How  many  bushels  of  corn  will  it  produce  at  32  bushels 
to  an  acre  ?     Find  the  value  of  the  crop  at  $  .48|  per 
bushel. 

9.  A  city  block  is  110  yards  long  by  90  yards  wide. 
How  many  acres  are  in  a  park  which  extends  7  blocks  one 
way  and  5  blocks  the  other  way  ? 

10.  A 'street  is  1,760  yards  long  and  20  yards  wide. 
How  many  thousand  bricks,  8  inches  by  4  inches,  will  be 
needed  to  pave  it  ? 

11.  How  many  square  tiles,  4  inches  on  a  side,  will  be 
required  to  tile  a  hall  60  feet  by  16  feet  ? 

12.  The  dimensions  of  a  room  are  16  feet  by  12  feet, 
and  10  feet  high.     How  many  square  yards   are  in  the 
four  walls  of  the  room  ?     How  many  square  yards  are  in 
the  walls  and  ceiling  ? 

13.  When  the  pressure  per  square  foot  of  a  hurricane 
is  19.47  pounds,  find  in  tons  the  total  pressure  exerted 
against  the  side  of  a  building  50  feet  long  45  feet  high. 


COMPUTATION  73 


COMPUTATION  ON  THE   BASIS   OF   100,   1,000,   AND   2,000 

Example  l.  Find  the  cost  of  transporting  5  bales  of 
cotton  weighing  respectively  510  lb.,  515  lb.,  508  lb., 
496  lb.,  487  lb.,  at  46^  per  100  lb. 

510  +  515  +  508  +  496  +  487  =  2516 

=  25.16  x  $  .46  =  $11.5736. 
Am.  $  11.57. 

Example  2.  Find  the  cost  of  shipping  7  head  of  cattle, 
average  weight  1,089  lb.,  at  97  ^  per  100  lb. 

7  *  J^89  x  $  .97  =  7  x  10.89  x  $  .97  =  $73.9431. 

Ana.  $73.94. 

10.89         The  shortest  way  to  multiply  by  .97  is 
7     to  take  .03  of  the  multiplicand  from  itself. 
76.23 

2.2869  =.03x76.23 
73.9431 

Example  3.  Find  the  value  of  a  car  load  of  coal  weigh- 
ing 43,275  lb.  at  $4.80  per  ton  of  2,000  lb. 


x  $4.80  =  x  $4.80  =  43.275  x$  2.40 

2000  2 

=  $103.86. 

Example  4.    How  much  will  it  cost  a  man  a  year  to  in- 
sure his  life  for  $8,750  if  the  annual  premium  is  $32.80 
per  1  1,000  ? 
8750 


1000 


x  $ 32.80  =  8.75  x  $ 32.80  =  $287.00. 


NOTE.    In  the  above  examples  the  sign  x  is  to  be  interpreted  as  mean- 
ing times. 


74  ADVANCED  BOOK  OF  ARITHMETIC 

EXERCISE  45 

The  following  rates  in  cents  per  100  Ib.  are  taken  from 
the  annual  Report  of  the  Railroad  Commission  of  the  state 
of  Texas  for  the  year  1906. 

Find  the  cost  of  shipping: 

1.  5  bales  cotton,  average  weight  503  Ib.,  @  45^. 

2.  12  bales  cotton,  average  weight  496  Ib.,  @  48^. 

3.  15  bales  cotton,  average  weight  490  Ib.,  @  8^. 

4.  130  bbl.  flour,  200  Ib.  to  the  barrel,  @  16  f. 

5.  124  bbl.  flour,  200  Ib.  to  the  barrel,  @  17  i. 

6.  1  carload  grain,  weighing  27,500  Ib.,  @  14^. 

7.  256  sacks  flour,  98  Ib.  to  the  sack,  @  12  £ 

8.  32,800  Ib.  grain  @  7j£ 

9.  1  carload  cotton  seed  products,  weighing  23,800  Ib., 


10.  1  carload  cotton  seed  hulls,  weighing  28,600  lb.,@ 

J* 

11.  1  carload   cotton  seed  meal,  weighing  42,000  Ib., 


@ 

12.  1  carload  cotton  seed  oil,  weighing  43,600  Ib.,  @  5^. 

13.  1  carload  brick,  weighing  45,000  Ib.,  @  5|  ^. 

14.  1  carload  fire  brick,  weighing  27,000  Ib.,  @  14|^. 

15.  1  carload  common   brick,  weighing  47,000  Ib.,  @ 

t. 

16.  1  carload  mules,  weighing  29,000  Ib.,  @  23^. 

17.  1  carload  cattle,  weighing  25,000  Ib.,  @  14^. 

18.  1  carload  sheep,  weighing  15,500  Ib.,  @  15/. 

19.  1  carload  crude  petroleum,  weighing  42,000  Ib.,  @ 


COMPUTATION  75 


20.  1  carload  asphaltum,  weighing  27,000  lb., 

21.  1  carload  melons,  weighing  20,500  lb.,  @ 

22.  5,880  lb.  molasses  @  7j£ 

23.  19,200  lb.  sugar  @  48^. 

24.  The  freight  rate  on  coal  in  cents  per  ton  of  2,000 
lb.  from  Eagle  Pass  to  the  points  named  is  : 

Weimer  138^  Flatonia  127^          Columbus  140^ 

Beaumont  217^          Gonzales  121^         Schulenburg  134^ 
Find  the  cost  of  shipping  1  carload  of  coal,  weighing 
39,000  lb.,  from  Eagle  Pass  to  each  of  these  points. 

25.  Find  the  cost  of  shipping  105,000  lb.  gravel  from 
Austin  to  San  Antonio  at  60^  per  ton  of  2,000  lb. 

26.  Find  the  cost  of  shipping  116,000  lb.  crushed  rock 
from  Clay  Quarry  to  Houston  at  67-|  ^  per  ton  of  2,000  lb. 

27.  Find  the  cost  of  shipping  130,000  lb.  crushed  rock 
from  Jacksboro  to  Fort  Worth  at  50  /  per  ton  of  2,000  lb. 

28.  Find  the  cost  of  shipping  a  carload  of  sand,  weigh- 
ing 50,000  lb.,  from  Kingsbury  to  San  Antonio  at  40^  per 
ton  of  2,000  lb. 

29.  Find  the  premium  on  a  $5,500  life  insurance  policy 
at  $21.50  per  11,000. 

30.  Find  the  premium  on  a  life  insurance   policy  for 
$4,500  at  $25.30  per  $  1,000. 

31.  What  is  the  premium  on  a  life  insurance  policy  of 
$6,500  at  $19.92  per  $1,000  ? 

32.  Find  the  premium  on  a  life  insurance  policy  for 
$10,500  at  $29.80  per  $1,000. 

33.  Find  the  premium   on  a  life  insurance  policy  for 
$8,500  at  $51.20  per  $1,000. 

34.  A    man    insured   his  life    for   $9,450.     Find    the 
annual  premium  at  $62.40  per  $1,000. 


76  ADVANCED   BOOK  OF  ARITHMETIC 

PERCENTAGE 

Per  cent  means  by  the  100,  or  on  the  100. 
Thus,  6  per  cent  means  6  on  100,  or  6  out  of  100. 
25  per  cent  means  25  on  100,  or  25  out  of  100. 
6  per  cent  is  written  6%;  25  per  cent  is  written  25%. 
If  a  man  invests  $100  and  gains  on  his  investment  $100, 
he  makes  a  profit  of  100%.    Therefore, 

100%  of  a  number  =  the  number. 
50  %  of  a  number  =  J  of  the  number. 
25  %  of  a  number  =  ^  of  the  number. 
20  %  of  a  number  =  ^  of  the  number. 
16|  %  of  a  number  =  ^  of  the  number. 
7  %  of  a  number  =  ^fa  of  the  number. 

The  following  per  cent  equivalents  should  be  remem- 
bered : 

100  %  =  1         50   %  =  1         331  %  =  1 

20    %  =  |  40    %  =  f         60   %  =  f 

80   %=|  16|%  =  i 

12-|  <j0  =  i  37-1  ^  =  3 

87i%=|  150   %  =  1|       175%=1| 

Example  l.  In  a  city  school  system  there  are  5250 
children  in  attendance.  If  84  %  are  promoted,  how  many 
are  promoted  ? 

5250  x  j^r  =  5250  x  .84  =  4410. 


Example  2.  Find  58J  %  of  3880. 
581  =  175  =  7 
100   300  12' 
3880  x  ^  =  22631. 


PERCENTAGE  77 

EXERCISE  46 
Find: 

1.  9  %  of  $  84.  14.  7  %  of  $1250. 

2.  8%  of  1425.  15.  25%  of  $4840. 

3.  6  %  of  $800.  16.  30  %  of  $3290. 

4.  5%  of  $2000.  17.  40%  of  $4500. 

5.  8%  of  $3250.  18.  50%  of  $3250. 

6.  7%  of  $4500.  19.  75%  of  $4000. 

7.  10%  of  $2250.  20.  70%  of  $3500. 

8.  11%  of  $4000.  21.  80%  of  $2450. 

9.  12%  of  $7250.  22.  100%  of  $7800. 

10.  4%  of  $3600.  23.  16%  of  $3200. 

11.  5%  of  $983.  24.  18%  of  $9200. 

12.  8%  of  $750.  25.  125%  of  $4000. 

13.  6  %  of  $850.  26.  225  %  of  $5400. 

27.  A  man  whose  salary  is  $750  a  year  saves  35%  of 
it.     How  much  does  he  save  ? 

28.  In  a  city  school  system  there  are  8250  children  ; 
54%  of  this  number  are  girls.     How  many  girls  are  in 
these  schools?     How  many  boys? 

29.  A  farm  of  175  acres  has  24%   woodland.      How 
many  acres  of  woodland  are  in  the  farm? 

30.  A  house  costs  $4740.     The  lot  on  which  it  is  built 
cost  32  %  of  the  value  of  the  house.     Find  the  cost  of  the 
lot. 

31.  In  a  certain  year  the  number  of  rainy  days  was 
20  %   of  the  number  of   days  in  the  year.     How  many 
rainy  days  were  there?     How  many  fair  days? 

32.  A  lawyer  charged  6  %  for  collecting  a  debt  of  $  3720. 
Find  his  fee.     How  much  did  he  remit  to  his  client? 


78  ADVANCED   BOOK  OF  ARITHMETIC 

EXERCISE  47 
Find: 

1.  331%  of  $9600.  10.  81%  of  $5640. 

2.  66f%of$3240.  11.  411%  of  $9120. 

3.  25%  of  $4920.  12.  58|%  of  $7560. 

4.  20%  of  $4500.  is.  6f%of$4515. 

5.  16-f%of$636.  14.  131%  of  $4845. 

6.  831%  of  $792.  15.  26f%of$3900. 

7.  12|%  of  $3280.  is.  46|%of$2400. 

8.  37|%  of  $4640.  17.  5f%of$3600. 

9.  621%  of  $5720.  is.  116|%of$672. 

19.  A  man  sells  his  house  for  $1800.     If  he  paid  for 
it  83^  %  of  the  price  at  which  it  was  sold,  what  did  the 
house  cost  ? 

20.  A  shoe  dealer  sold  $720  worth  of  shoes.     The  shoes 
cost  him  66|%  of  the  selling  price.     Find  the  cost  price 
of  the  shoes. 

21.  In  an  apple  orchard  of  840  trees  58|  %  bore  fruit. 
How  many  trees  were  fruit- bearing  ? 

22.  A  ranchman  lost  during  a  blizzard  16|%   of  his 
sheep.     If  the  number  in  his  flock  was  originally  960,  how 
many  did  he  lose,  and  how  many  were  left  ? 

23.  If  the  area  of  a  county  is  1230  square  miles,  and  75  % 
of  it  arable  land,  how  many  square  miles  are  arable  land  ? 

24.  Piles  used  in  the  construction  of  a  railroad  bridge 
are  42  ft.  long,  and  83|  %  of  their  length  is  beneath  the 
water.     Find  the  length  in  the  water. 

25.  The  railroad  mileage  of  the  United  States  in  the  year 
1904  was  212,578.     Of  this  the  railroad  mileage  of  Florida 
was  1|  %.      Find  the  railroad  mileage  of  Florida  in  1901. 


INTEREST  AND  PROPERTY   INSURANCE  79 


INTEREST   AND   PROPERTY  INSURANCE 

Interest  is  money  paid  for  the  use  of  money. 

The  sum  loaned  is  the  principal. 

Interest  is  always  reckoned  as  a  rate  per  cent  of 
the  principal.  The  rate  per  cent  is  for  one  year  unless 
otherwise  stated. 

Property  insurance  is  idemnity  against  loss  of  property 
and  is  reckoned  as  a  rate  on  the  basis  of  $100  valuation. 

The  sum  paid  for  insurance  is  the  premium. 

The  written  contract  between  the  assured  and  the 
insurance  company  is  called  the  insurance  policy. 

Example.  What  is  the  premium  on  an  insurance 
policy  of  $15,350  at  $1.35  per  $100? 

x  $1.35  =  153.5  x  $1.35  =  $207.225,  or 
15350  x  $.0135  =  $207.225. 

In  the  first  solution,  the  number  of  100\s  is  multiplied 
by  the  rate  on  $100.  In  the  second  solution,  the  number 
of  dollars  is  multiplied  by  the  rate  on  $1.00. 

EXERCISE  48 
Find  the  interest  on: 

1.  $600  for  1  yr.  at  4%;  for  1  yr.  at  5%;  for  1  yr.  at 
6%;  for  1  yr.  at  8%. 

2.  $850  for  1  yr.  at  7%;  for  1  yr.   at  8%;  for  1  yr. 
at  9%. 

3.  $950  for  1  yr.  at  3%;  for  1  yr.  at  4%;  for  1  yr. 

at  S%. 

4.  $982  for  2  yr.  at  4%;  for  3  yr.  at  5%;  for  1  yr. 
6  mo.  at  6%. 


80  ADVANCED   BOOK  OF  ARITHMETIC 

5.  $738  for  £  yr.  at  5  % ;  J  yr.  at  6  %. 

6.  $920  for  \  yr.  at  6  % ;  i  yr.  at  7  %. 

7.  $1200  for  4  mo.  at  5%;  4  mo.  at  6%. 

8.  $1100  for  6  mo.  at  7%;  6  mo.  at  4%. 

9.  $1280  for  3  mo.  at  8%;   3  mo.  at  6%. 

Find  the  premium  for  insuring  dwellings  against  loss 
by  fire  at  the  rates  specified  per  $100: 

10.  $2500  at  $1.30.  26.  $22,500  at  $1.10. 

11.  $2000  at  $1.15.  27.  $18,250  at  $1.20. 

12.  $4500  at  $1.50.  28.  $2400  at  $1.80. 

13.  $3000  at  $1.25.  29.  $9300  at  $1.50. 

14.  $2500  at  $1.90.  30.  $8500  at  $1.60. 

15.  $5500  at  $1.70.  31.  $9450  at  $1.50. 

16.  $6500  at  $1.50.  32.  $6500  at  $1.90. 

17.  $4000  at  $1.80.  33.  $5400  at  $1.60. 

18.  $5400  at  $1.70.  34.  $9500  at  $1.90. 

19.  $3300  at  $1.80.  35.  $12,000  at  $1.75. 

20.  $7500  at  $1.60.  36.  $18,000  at  $1.75. 

21.  $7250  at  $1.40.  37.  $200,000  at  $1.25. 

22.  $10,500  at  $1.30.  38.  $15,500  at  $1.60. 

23.  $19,250  at  $1.25.  39.  $16,200  at  $1.60. 

24.  $16,450  at  $1.60.  40.  $1800  at  $1.90. 

25.  $7900  at  $1.35.  41.  $1750  at  $1.75. 

42.  A  man  insures  his  residence,  valued  at  $5000,  at  | 
of  its  value  at  the  rate  of  $1.20  on  the  $100.    Find  the 
premium  paid. 

43.  A  jobber  insures  a  quantity  of  cotton,  worth  $  30,000, 
at  |  of  its  value  at  the  rate  of  75^  on  the  $100.     Find  his 
premium. 


CHAPTER   II 

COMPOUND   QUANTITIES 

CONCRETE  quantities  of  the  same  kind,  but  consisting 
of  units  of  different  denominations,  are  called  compound 
quantities. 

Seventeen  days,  10  hours,  and  30  minutes  is  a  compound 
quantity.  Here,  we  have  three  units  of  measurement; 
namely,  a  day,  an  hour,  and  a  minute.  These  units  are  of 
different  denominations,  but  each  is  of  the  same  kind, 
inasmuch  as  it  stands  for  a  definite  portion  of  time. 

Compound  quantities  are  also  called  compound  denominate 
quantities.  Quantities  composed  of  units  of  one  denomi- 
nation are  generally  called  simple  quantities. 

AVOIRDUPOIS  WEIGHT 

16  ounces  (oz.)  =  1  pound  (Ib.) 

100  pounds  =  1  hundredweight  (cwt.) 
20  hundredweight,  or  2000  pounds  =  1  ton  (T.) 
2240  pounds  =  1  long  ton 

Avoirdupois  Weight  is  used  in  weighing  all  commercial 
quantities  excepting  the  precious  metals,  jewels,  and  drugs, 
when  sold  by  retail  druggists. 

The  unit  in  Avoirdupois  Weight  is  the  pound  of  7000 
grains.  One  cubic  inch  of  distilled  water  weighs  in  vacuo 
252.286  grains,  of  which  7000  weigh  1  pound. 

The  long  ton  is  used  in  the  United  States  custom 
houses,  and  in  weighing  coal  and  mineral  products  at  the 
mines. 

G  81 


82  ADVANCED   BOOK  OF  ARITHMETIC 

The  process  of  reducing  units  of  any  given  denomi- 
nation to  units  of  higher  denomination  is  called  reduction 
ascending. 

The  process  of  reducing  units  of  a  higher  denomination 
or  of  higher  denominations  to  units  of  lower  denomination 
is  called  reduction  descending. 

Example.    Reduce  1,000,201  oz.  to  higher  denominations. 

Divide  by  16  to  get  the  num- 

16 1000201 ber  of  pounds.     Divide  by  100 

100 

20 


62512  Ib.  9  oz.          to  get  the  number  of  hundred- 


625  cwt.  12  Ib.      weights.     Divide  by  20  to  get 


31  T.  5  cwt.        the  number  of  tons.     Ans.   31 

T.  5  cwt.  12  Ib.  9  oz. 
Weights  are  generally  expressed  in  tons  or  in  pounds. 

EXERCISE  49 

Reduce  to  higher  denominations  : 

1.  7800  oz.  3.    75,497  oz.  5.    7987  Ib. 

2.  9763  oz.  4.    1,000,000  oz.  6.    32,721  Ib. 

7.  How  many  ordinary  or  short  tons  in  100  long  tons  ? 

8.  Reduce  10,000  Ib.  to  long  tons. 

Example.     Reduce  7  T.  3  cwt.  12  Ib.  10  oz.  to  ounces. 

T.     CWT.     LB.        OZ. 

7     3     12  10 

20 

143  cwt.  Reduce  the  7  T.  to   hundredweights  by 

100  multiplying  by  20.     Add  3  cwt.  to  the  prod- 

14312  Ib.  uct  and  get  143  cwt.     Multiply  143  by  100 

16  and  add  12  Ib.  to  the  product.     This  gives 

85882  14,312  Ib.     Multiply  this  by  16,  adding  10 

14312  oz.,  when  the  first  figure  is  multiplied  by  6. 
2  29002  oz. 


COMPOUND  QUANTITIES  83 

EXERCISE   50 

1.  Reduce  19  T.  to  pounds. 

2.  Reduce  14  T.  4  cwt.  to  pounds. 

3.  Reduce  17  T.  3  cwt.  to  pounds. 

4.  Reduce  25  T.  2  cwt.  to  ounces. 

5.  Reduce  3  T.  15  cwt.  2  Ib.  to  pounds. 

6.  Reduce  4  T.  11  cwt.  58  Ib.  to  pounds. 

7.  Reduce  8  T.  2  cwt.  73  Ib.  to  pounds. 

8.  A  dealer  buys  50  long  tons  of  coal  and  sells  it  by 
the  short  ton.     How  many  short  tons  does  he  sell? 

9.  A  dealer  buys  100  long  tons  of  coal  at  $6.75  per 
ton.     He  sells  it  by  the  short  ton  at  $6.75  per  ton.     How 
much  profit  does  he  make  ? 

10.  Convert  784  short  tons  into  long  tons. 

11.  Convert  550  long  tons  into  short  tons. 

12.  Three   horses    together   weigh    2  T.   4  cwt.  91  Ib. 
Find  in  pounds  the  average  weight  of  the  horses. 

LINEAR  OR  LONG  MEASURE 
12  inches  (in.)     =  1  foot  (ft.) 
3  feet  =  1  yard  (yd.) 

5£  yards  =  1  rod  (rd.) 

320  rods  =  1  mile  (mi.) 

6080  feet  =  1  knot,  geographical  or  nautical  mile 

3  knots  =  1  marine  league 

1  mi.  =  320  rd.  =  1760  yd.  =  5280  ft. 

The  unit  of  length  is  the  yard. 

The  yard  in  the  United  States  is  denned  as  f  f $$  of  the 
meter. 

The  standard  yard  of  this  country  has  been  adopted 
since  1893. 


84  ADVANCED  BOOK  OF  ARITHMETIC 

EXERCISE  51 

1.  Reduce  4  yd.  2  ft.  to  inches. 

2.  Reduce  110  yd.  1  ft.  to  inches. 

3.  Reduce  5J  mi.  to  yards. 

4.  Reduce  7  mi.  120  rd.  to  yards. 

5.  Reduce  10  mi.  110  rd.  4  yd.  to  yards. 

6.  Reduce  445|  mi.  to  yards. 

7.  Reduce  7.74  mi.  to  yards. 

8.  Reduce  8.35  mi.  to  rods. 

9.  Reduce  238  rd.  to  feet. 

SQUARE    MEASURE 

144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.) 

9  square  feet  =  1  square  yard  (sq.  yd.) 

30£  square  yards  =  1  square  rod  (sq.  rd.) 

160  square  rods  =  1  acre  (A.) 

640  acres  =  1  square  mile  (sq.  mi.) 

1  acre  =  4840  square  yards 

1  section  =  1  square  mile 

36  sections  =  1  township 

Square  measure  is  used  to  measure  the  areas  of  surfaces. 

A  cube  is  a  solid  bounded  by  six  plane  surfaces,  each  of 
which  is  a  square. 

A  solid  having  the  shape  of  a  box  or  of  an  ordinary 
room,  i.e.  one  bounded  by  six  plane  surfaces,  each  of  which 
is  a  rectangle,  is  called  a  rectangular  solid. 

The  volume  of  a  solid  means  the  amount  of  space  it 
occupies.  This  is  measured  by  the  number  of  times  the 
solid  contains  the  unit  of  measurement. 

The  unit  of  volume  is  a  cube  having  for  an  edge  the 
linear  unit.  The  cubic  unit  from  which  all  others  are 
derived  is  the  cubic  yard. 


COMPOUND  QUANTITIES  85 

CUBIC   OR  SOLID   MEASURE 

1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (cu.  ft.) 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.) 

MEASURES  OF   CAPACITY 

There   are.  two  measures   of   capacity   in   general   use; 
namely,  Liquid  Measure  and  Dry  Measure. 

LIQUID  MEASURE 

4  gills  (gi.)  =  1  pint  (pt.) 
2  pints  =  1  quart  (qt.) 

4  quarts        =  1  gallon  (gal.) 
A  gallon  contains  231  cu.  in. 

DRY  MEASURE 

2  pints  (pt.)  =  1  quart  (qt.) 
8  quarts  =  1  peck  (pk.) 
4  pecks  =  1  bushel  (bu.) 

One  bushel  contains  2150.42  cu.  in.     It  is  the  volume 
of  a  cylindrical  vessel  18 J  in.  in  diameter  and  8  in.  deep. 

REDUCTION  DESCENDING 

Example.    Reduce  5  gal.  2  qt.  1  pt.  2  gi.  to  gills. 
5  gal.  2  qt.  1  pt.  2  gi. 
A 
20  =  number  of  quarts  in  5  gal. 

2 

22  =  number  of  quarts  in  5  gal.  2  qt. 
'     _2 

44  =  number  of  pints  in  5  gal.  2  qt. 
_1 

45  =  number  of  pints  in  5  gal.  2  qt.  1  pt. 
4 

182  =  number  of  gills  in  5  gal.  2  qt.  1  pt.  2  gi. 


86  ADVANCED  BOOK  OF  ARITHMETIC 

EXERCISE   52 

Reduce  : 

1.  1  sq.  mi.  to  sq.  rd.  14.    20.25  cu.  yd.  to  cu.  in. 

2.  2|  sq.  mi.  to  A.  15.    2  gal.  2  qt.  to  qt. 

3.  12  A.  to  sq.  ft.  16.    5  gal.  3  qt.  to  qt. 

4.  27  sq.  rd.  to  sq.  ft.  17.    3  gal.  1  pt.  to  pt. 

5.  3  mi.  50  rd.  to  ft.  18.    7  gal.  1  pt.  to  pt. 

6.  8  mi.  40  rd.  to  ft.  19.    19,25  gal.  to  pt. 

7.  2|  mi.  to  yd.  20.   4  bu.  to  qt. 

8.  3.75  mi.  to  yd.  21.    3|  bu.  to  qt. 

9.  2.125  mi.  to  ft.  22.    3.625  bu.  to  qt. 

10.  25  cu.  yd.  to  cu.  ft.  23.    7  pk.  to  qt. 

11.  38  cu.  yd.  20  cu.  ft.  to  cu.  ft.  24.    7.375  pk.  to  pt. 

12.  171  cu.  yd.  to  cu.  ft.  25.    18J  bu.  to  pt. 

13.  18.75  cu.  yd.  to  cu.  ft.  26.    13  bu.  3  qt.  to  pt. 

27.  How  many  feet  are  in  |  mi.  ?  in  |  mi.  ?  in  -fa  mi.  ? 

28.  How  many  yards  are  in  ^  mi.  ?  in  ^  mi.  ?  in  J  mi.  ? 

29.  What  fraction  of  a  mile  is  440  yd.  ?  176  yd.  ?  88  yd.  ? 

30.  How  many  square  yards  are  in  ^  of  an  A.  ?  in  J  A.  ? 

31.  What  part  of  a  township  is  1  sq.  mi.  ? 

32.  How  many  square  rods  are  in  £  A.  ? 

33.  How  many  square  rods  are  in  .7  A.  ?  in.  .9  A.  ? 

34.  How  many  square  feet  are  in  f  sq.  rd.  ? 

35.  How  many  cubic  inches  are  in  1  pt.,  Dry  Measure  ? 

36.  How  many  cubic  inches  are  in  1  pt.,  Liquid  Measure? 

37.  How  many  quarts  are  in  f  pk.  ? 

38.  How  many  gallons  are  required  to  fill  10  bu.  measures? 


COMPOUND   QUANTITIES  87 


REDUCTION  ASCENDING 

Example  l.    Reduce  85  pt.  to  higher  denominations. 
85  pt. 


42  qt.    1  pt. 


10  gal.  2  qt.  1  pt. 
Example  2.    The  length  of  one  degree  of  latitude  at 
40°  north  is  364,280  ft.     Express  this  length  in  miles. 

There  are  5280  ft.  in  1  mi.     The  fac- 


80 

6 

11 


364280  ft. 


4553  g  '  tors  of  5280  are  80,  6,  and  11.  (A  num- 
ber is  divided  by  80  by  dividing  by  8 
and  writing  each  quotient  figure  one 
place  to  the  right.)  Ans.  68.992  mi. 


EXERCISE   53 

Reduce  to  higher  denominations: 

1.  234  pt.  (Liquid  Measure).  5.    2000 pt.  (Dry  Measure). 

2.  47,385  cu.  in.  6.    393,000  cu.  in. 

3.  3456  pt.  (Liquid  Measure).  7.    20,000,000  A. 

4.  10,240  rd.  8.    15,000  sq.  in. 

9.  The  equatorial  diameter  of  the  earth  is  41,852,404  ft. 
Express  this  distance  in  miles  and  the  decimal  of  a  mile 
correct  to  two  decimal  figures. 

10.  The  polar  diameter  of  the  earth  is  41,709,790  ft. 
What  is  the  polar  diameter  of  the  earth  in  miles  correct 
to  two  decimal  figures  ? 

11.  By  how  many  miles  does  the  equatorial  diameter 
exceed  the  polar  diameter  ? 

12.  Light  takes  8  min.  18  sec.  to  come  from  the  sun  to 
the  earth.     The  mean  distance  of  the  sun  from  the  earth 
is  92,790,000  mi.     Find  the  velocity  of  light  per  second. 


88 


ADVANCED   BOOK  OF  ARITHMETIC 


CIRCULAR   ARC   MEASURE 

A  circle  is  a  plane  figure  bounded  by  a  line  called  the 
circumference,  every  point  of  which  is  equally  distant  from 
a  point  within  the  figure  called  the  center. 

A  straight  line  from  the 
center  to  the  circumference 
is  called  a  radius. 

A  straight  line  drawn 
through  the  center  and  ter- 
minated by  the  circumference 
is  called  a  diameter. 

The  lines  AB  and  CD  are 
diameters. 

Any  portion  of  a  circum- 
ference is  called  an  arc.  mn 
is  an  arc. 

An  arc  equal  to  one  half  of  a  circumference  is  called  a 
semicircumference. 

An  arc  equal  to  one  fourth  of  a  circumference  is  called 

a  quadrant. 

60  seconds  (")  =  1  minute  (') 
60  minutes        =  1  degree  (°) 
360  degrees         =  1  circumference 

ANGULAR  MEASURE 

An  angle  is  a  figure  formed  by  two  straight  lines  pro- 
ceeding from  a  point.  Its  magnitude  depends  upon  the 
amount  of  turning  necessary  to  bring  one  side  into  co- 
incidence with  the  other. 

NOTE.  Beginners  should  be  provided  with  a  protractor  and  they 
should  draw  and  measure  angles.  To  learn  things  by  actual  trial  and  not 
by  mere  hearsay  is  to  educate. 


COMPOUND  QUANTITIES 


89 


If  one  straight  line  meets  another  straight  line  so  as  to 
make  the  adjacent  angles  equal  to  each  other,  each  angle 

is  called  a  right  angle. 

S 


If  the  lines  MN,  ST  meet  in  B  so  as  to  make  the  angles 

,  SBM  equal,  then  each  angle  is  a  right  angle. 
The  unit  of  angular  measure  is  1  degree. 

60  seconds  (")  =  1  minute  (') 
60  minutes        =  1  degree  (°) 
90  degrees         —  1  right  angle 
2  right  angles  =  1  straight  angle 

EXERCISE  54 

1.  What  part  of  V  is  1"  ? 

2.  How  many  seconds  are  in  |  minute  ? 

3.  Reduce  1'  30"  to  seconds. 

4.  What  part  of  a  straight  angle  is  a  right  angle  ? 

5.  What  part  of  a  right  angle  is  an  angle  of  45°  ?  30°  ? 
15°?  18°?  60°?  75°? 

6.  How  many  degrees  are  in  1  straight  angle  ?  in  |  of 
a  straight  angle  ? 

7.  What  part  of  a  straight  angle  is  an  angle  of  15°  ?  24°  ? 
30°  ?  45°  ?  60°  ?  80°  ?  1006  ?  105°  ?  120°  ?  135°  ?  150°  ? 


90  ADVANCED  BOOK  OF  ARITHMETIC 

TIME  MEASURE 

60  seconds  (sec.)  =  1  minute  (min.) 

60  minutes  =  1  hour  (hr.) 

24  hours  =  1  day  (da.) 

7  days  =  1  week  (wk.) 

365  days  =  1  common  year  (yr.) 

366  days  —  1  leap  year  (yr.) 
100  years  =  1  century 

There  are  twelve  calendar  months  in  a  year. 
The  following  lines  will  enable  one  to  remember  the 
number  of  days  in  each  month  : 

"  Thirty  days  hath  September, 
April,  June,  and  November, 
February  twenty-eight  alone, 
And  all  the  others  thirty-one ; 
But  leap  year,  coming  once  in  four, 
Gives  February  one  day  more." 

A  day  is  the  standard  unit  of  time.  It  is  of  the  same 
duration  at  all  places.  It  represents  the  period  of  time 
that  elapses  between  two  successive  passages  of  the  sun 
across  the  meridian  of  any  place. 

The  length  of  a  year  is  365  daj^s,  5  hours,  48  minutes, 
46  seconds.  The  common  year  has  365  days.  The  differ- 
ence in  length  between  the  common  year  and  the  actual, 
or  solar  year,  gave  rise  to  the  introduction  of  leap  years. 
Centennial  years  are  leap  years  when  the  number  of  the 
year  is  exactly  divisible  by  400.  Thus,  the  year  2000  is 
a  leap  year  because  2000  is  divisible  by  400.  All  other 
years  are  leap  years  when  their  numbers  are  exactly 
divisible  by  4.  The  year  1907  is  not  a  leap  year,  as  the 
number  1907  is  not  exactly  divisible  by  4.  The  year 
1828  was  a  leap  year,  as  1828  is  exactly  divisible  by  -4. 


COMPOUND   QUANTITIES  91 

MISCELLANEOUS  MEASURE 


1  bushel  of  barley 

=  48  Ib. 

1  bushel  of  wheat 

=  60  Ib. 

1  bushel  of  oats 

=  32  Ib. 

1  bushel  of  rye 

=  56  Ib. 

1  bushel  of  potatoes  (Irish) 

=  60  Ib. 

1  bushel  of  potatoes  (sweet) 

=  55  Ib. 

1  bushel  of  buckwheat 

=  48  Ib. 

1  bushel  of  beans 

=  60  Ib. 

1  bushel  of  shelled  corn 

=  56  Ib. 

1  bushel  of  peas 

=  60  Ib. 

1  bushel  of  clover  seed 

=  60  Ib. 

1  barrel  of  flour 

=  196  Ib. 

1  barrel  of  pork  or  beef 

=  200  Ib. 

1  cental  of  grain 

=  100  Ib. 

NUMBERS  PAPER  MEASURE 

12  units  =  1  dozen  (doz.)  24  sheets  of  paper  =  1  quire 

12  dozen  =  1  gross  20  quires  =  1  ream 

12  gross  =  1  great  gross  2  reams  =  1  bundle 

20  units  =  1  score  5  bundles  =  1  bale 

TROY   WEIGHT 

24  grains  (gr.)     =  1  pennyweight  (pwt.) 
20  pennyweights  =  1  ounce  (oz.) 
12  ounces  =  1  pound  (Ib.) 

1  pound  Troy       =  5760  grains 

Troy  weight  is  used  in  weighing  precious  metals  and 
jewelry. 

(The  measures  in  this  paragraph  are  inserted  merely  for 
reference.) 


92  ADVANCED   BOOK  OF  ARITHMETIC 

EXERCISE  55 

Reduce  to  seconds : 

1.  18°  20'  20".         3.    12°  5'  10".         5.    120.3°. 

2.  A  quadrant.         4.    7|°.  6.    45°  30'  20". 

Reduce  to  minutes: 

7.  14i°.  9.    254.125°.          11.    18f°. 

8.  75.75°.  10.    4f°.  12.   13|°. 

13.  Reduce  a  common  year  to  minutes. 

14.  Find  the  number  of  minutes  in  the   years  1903, 
1904,  1905. 

15.  Find  the  number  of  minutes  in  February,  1904. 

16.  Find   the   number   of   minutes    in  the  first   three 
months  of  the  year  1903 ;  also  in  the  first  three  months  of 
the  year  1904. 

17.  Find  the  number  of  seconds  in  a  solar  year,  con- 
sisting of  365  da.  5  hr.  48  min.  46  sec. 

18.  The   pulse  of  a  healthy  person   beats  70  times  a 
minute.     At  this  rate,  how  many  times  will  it  beat  in  a 
leap  year  ?     How  many  times  will  it  beat  in  the  four  suc- 
cessive years,  beginning  1904? 

19.  Reduce  30  wk.  6  da.  12  hr.  to  minutes. 

20.  Reduce  1|  common  years  to  days. 

21.  Reduce  5f  wk.  to  hours. 

22.  Reduce  20.4  yr.  to  hours,  allowing  five  of  them  to 
be  leap  years. 

23.  How  many  days  are  there  between  Jan.  30,  1902, 
and  Jan.  30,  1910  ? 

24.  Reduce  T2^  of  a  circumference  to  degrees. 

25.  Reduce  ||-  of  a  straight  angle  to  degrees. 


COMPOUND   QUANTITIES  93 

ADDITION 

In  adding  compound  quantities,  proceed  as  follows  : 

Step  1.  Arrange  the  quantities  so  that  units  of  the 
same  denomination  stand  in  the  same  vertical  column,  the 
highest  denomination  being  written  first,  the  next  to 
the  highest  second,  and  so  on. 

Step  2.  Beginning  with  the  right-hand  column,  add  the 
numbers  in  it,  divide  their  sum  by  the  number  of  units 
which  makes  one  unit  of  the  next  higher  denomination. 
Write  the  remainder  in  the  right-hand  column,  and  carry 
the  quotient  to  the  next  column. 

Step  3.  Treat  the  next  column  to  the  left  in  the  same 
manner.  The  remaining  columns  are  dealt  with  in  the 
same  way. 

LINEAR  MEASURE 
EXERCISE  56 

Add:        (1)  (2)  (3) 

YD.      FT.      IN.  YD.      FT.      IN.  YD.      FT.      IN. 

908  26  16  51  11 

11     2     4  33  2     6  312 

61  10  20  10  413 

506  70  1     9  11     2    5 

SQUARE  MEASURE 

(*)  (5)  (6) 

A.      SQ.  BD.  A.      SQ.  BD.  A.      SQ.  BD. 

76  144  33  79  127  38 

85  131  173  27  192  99 

37  33  254  28  238  77 

63  99  45  53  413  25 


94  ADVANCED  BOOK  OF  ARITHMETIC 

CAPACITY 

7.  Add:  2  gal.  3  qt.  1  pt.,  3  gal.  2  qt.  1  pt.,  5  gal.  2 
qt.  1  pt.,  4  gal.  2  qt.  1  pt. 

8.  Add:  7  gal.  2  qt.  1  pt.,  9  gal.  3  qt.,  4  gal.  1  qt.  1 
pt.,  6  gal.  3  qt.  1  pt.,  9  gal.  1  pt.,  7  gal.  1  pt. 

9.  Add:  3  bu.  3  pk.  5  qt.,  4  bu.  2  pk.  4  qt.,  9  bu.  2 
pk.  7  qt.,  9  bu.  7  qt.,  8  bu.  2  pk.  3  qt.,  6  bu.  3  pk.  2  qt. 

10.  Add  :  4  bu.  7  qt.,  3  bu.  4  pk.  6  qt.,  7  bu.    2  pk.  6 
qt.,  8  bu.  3  qt.,  9  bu.  2  pk.  3  qt.,  4  bu.  3  pk.  2  qt. 

11.  Add :  17  gal.  1  pt.,  14  gal.  2  qt.  1  pt.,  2  gal.  2  qt.  1 
pt.,  15  gal.  1  pt.,  13  gal.  1  qt.  1  pt.,  14  gal.  3  qt.  1  pt. 

12.  Add:  14  bu.  2  pk.  7  qt.,  29  bu.  3  pk.  5  qt.,  23  bu. 

2  pk.    6  qt.,  39  bu.  6  qt.,   28  bu.  3  pk.,  17  bu.  2  pk.  5  qt. 

13.  Add  :  38  bu.  3  pk.  2  qt.,  16  bu.  2  pk.  1  qt.,  28  bu. 

3  pk.  7  qt.,  3  bu.  7  qt.,  5  bu.  3  pk.,  24  bu.  2  pk.  2  qt. 

14.  Add  :  15  bu.  5  qt.,  12  bu.  3  pk.,  17  bu.  7  qt.,  18  bu. 
6  qt.,  29  bu.  2  pk.  3  qt.,  71  bu.  3  pk.  2  qt.,  18  bu.  3  pk. 

AVOIRDUPOIS   WEIGHT 

is.    Add:  20  T.  215  lb.,  18  T.  425  lb.,  17  T.   328   lb., 
92  T.  411  lb. 

16.  Add:  384  lb.  12  oz.,  125  lb.  15  oz.,  82  lb.  14  oz., 
73  lb.  11  oz. 

17.  Add:  425  lb.  10  oz.,  17  lb.  14  oz.,  30  lb.  12  oz.,  72 
lb.  9  oz. 

18.  Add:  15  T.  290  lb.,  17  T.  184  lb.,  12  T.  127  lb., 
15  T.  9  lb.,  18  T.  18  lb. 

19.  Add  :  18  lb.  8  oz.,  64  lb.  7  oz.,  82  lb.  6  oz.,  90  lb. 
5  oz.,  16  lb.  13  oz. 

20.  Add:  16  T.  175  lb.,  71  T.  29  lb.,  28  T,  245  lb., 
97  T.  159  lb.,  13  T.  1300  lb. 


COMPOUND   QUANTITIES  95 

TIME 

21.  Add:  5  da.  4  hr.  15  min.,  17  da.  17  hr.  17  min., 
92  da.  14  hr.  14  min.,  27  da.  23  hr.  12  min.,  29  da.  16  hr. 
14  min.,  45  da.  15  hr.  18  min. 

22.  Add  :  4  wk.  5  da.  7  hr.,  9  wk.  6  da.  11  hr.,  18  wk. 

5  da.  12  hr.,  23  wk.  11  hr.,  28  wk.  4  da.  4  hr.,  73  wk. 

6  da.  19  hr.,  82  wk.  5  da.  21  hr. 

23.  Add :  20  hr.  30  min.  18  sec.,  17  hr.  45  min.  37  sec., 
14  hr.  18  min.  18  sec.,  14  hr.  12  min.  12  sec.,  9  hr.  48  min. 

48  sec.,  8  hr.  39  min.  39  sec. 

24.  Add:  12  da.  17  hr.  44  min.,  15  da.  18  hr.  18  min., 
31  da.  19  hr.  19  min.,  33  da.  21  hr.  27  min.,  12  da.  12  hr. 
36  min.,  34  da.  20  hr.  23  min. 

25.  Add:  3  wk.  5  da.  23  hr.,  8  wk.  6  da.  16  hr.,  9  wk. 
5  da.  18  hr.,  4  wk.  4  da.  14  hr.,  10  wk.  5  da.  13  hr. 

26.  Add  :  14  hr.  14  min.  14  sec.,  9  hr.  54  min.  38  sec., 
11  hr.  12  min.  19  sec.,  4  hr.  31  min.  27  sec.,  5  hr.  45  min. 
43  sec.,  8  hr.  41  min.  42  sec. 

VOLUME 

27.  Add  :  4  cu.  ft.  1421  cu.  in.,  9  cu.  ft.  294  cu.  in., 
18  cu.  ft.  998  cu.  in.,  7  cu.  ft.  778  cu.  in.,  9  cu.  ft.  499 
cu.  in.,  15  cu.  ft.  498  cu.  in. 

28.  Add:  27  cu.  yd.  19  cu.  ft.,  84  cu.  yd.  24  cu.  ft., 
87  cu.  yd.  19  cu.  ft.,  16  cu.  yd.  22  cu.  ft.,  55  cu.  yd. 
17  cu.  ft.,  34  cu.  yd.  16  cu.  ft. 

29.  Add  :  37  cu.  yd.  13  cu.  ft.,  38  cu.  yd.  26  cu.  ft., 

49  cu.  yd.  25  cu.  ft.,  62  cu.  yd.  26  cu.  ft.,  77  cu.  yd. 

17  cu.  ft.,  94  cu.  yd.  28  cu.  ft. 

30.  Add:  15  cu.  ft.  578  cu.  in.,  18  cu.  ft.  902  cu.  in., 

18  cu.  ft.  978  cu.  in.,  15  cu.  ft.  293  cu.  in. 


96  ADVANCED   BOOK   OB^   ARITHMETIC 

SUBTRACTION 

From  19  sq.  yd.  5  sq.  ft.  20  sq.  in.  take  14  sq.  yd. 
7  sq.  ft.  45  sq.  in. 

SQ.  YD.      SQ.  FT.      SQ.  IN. 

19        5        20 

14         7        45 

4        6       119 

Step  1.  Write  the  quantities  so  that  units  of  the  same 
denomination  are  in  the  same  column. 

Step  2.  Find  what  concrete  quantity  added  to  45  sq.  in. 
will  give  1  sq.  ft.  20  sq.  in.,  i.e.  164  sq.  in.  Write  the 
remainder,  119  sq.  in.,  in  the  column  for  square  inches. 
Carry  1  sq.  ft. 

Step  3.  Find  what  concrete  quantity  added  to  8  sq.  ft. 
will  give  1  sq.  yd.  5  sq.  ft.,  i.e.  14  sq.  ft.  Write  the  re- 
mainder, 6  sq.  ft.,  in  the  column  for  square  feet.  Carry 
1  sq.  yd. 

Step  4.  Find  the  difference  between  15  sq.  yd.  and  19 
sq.  yd.  and  write  it  in  its  proper  place. 

EXERCISE  57 
CIRCULAR   ARC   OR  ANGULAR  MEASURE 

1.  Subtract  5°  12'  13"  from  84°  14'  30". 

2.  Subtract  19°  14'  14"  from  27°  15'  10". 

3.  Subtract  38°  15'  45"  from  90°  10'  10". 

4.  Subtract  54°  14'  54"  from  172°  0'  19", 

5.  Subtract  84°  5'  15"  from  90°. 

6.  Subtract  113°  13'  54"  from  180°. 

7.  Subtract  94°  53'  50"  from  180°, 


COMPOUND  QUANTITIES  97 

8.  Subtract  87°  15'  from  133°  12'. 

9.  Subtract  119°  54'  17"  from  180°. 

10.  Subtract  15°  14'  17"  from  94°  14'  7". 

11.  The  tropic  of  Cancer  is  23°  27'  6"  north  of  the  equa- 
tor ;  the  Arctic  circle  is  23°  27'  6"  south  of  the  north  pole. 
Find  the  width  of  the  north  temperate  zone. 

CAPACITY 

12.  From  3  bu.  2  pk.  4  qt.  take  1  bu.  2  pk.  5  qt. 

13.  From  12  gal.  3  qt.  1  pt.  take  4  gal.  3  qt. 

14.  From  11  gal.  take  4  gal.  3  qt.  1  pt. 

15.  From  17  gal.  take  11  gal.  1  qt.  1  pt. 

16.  From  37  bu.  2  pk.  4  qt.  take  17  bu.  3  pk.  7  qt. 

17.  From  29  bu.  1  pk.  2  qt.  take  19  bu.  3  pk.  5  qt. 

18.  From  37  gal.  take  17  gal.  1  qt.  1  pt. 

19.  From  134  gal.  take  112  gal.  3  qt.  1  pt. 

20.  From  1J  gal.  take  f  gal.  and  express  the  result  in 
quarts. 

21.  From  If  bu.  take  |  bu.  and  express  the  result  in 
quarts. 

22.  From  5^-  bu.  take  If  bu.  and  express  the  result  in 
quarts. 

TIME 

23.  From  3  da.  4  hr.  11  min.  take  1  da.  7  hr.  14  min. 

24.  From  11  da.  5  hr.  10  min.  take  4  da.  11  hr.  19  min. 

25.  Almanacs    give    the    time   of    sunrise   in   Florida, 
Louisiana,   and   Texas    on  March   5   as  6.22   A.M.,   and 
that   of    sunset   as   6.2   P.M.     Find    the    length   of   the 
day. 


98  ADVANCED  BOOK  OF  ARITHMETIC 

26.  On  April  1,  1903,  the  moon  rose  at  10.28  P.M.  On 
April  4  following,  it  rose  at  12.28  A.M.  How  many 
hours  and  minutes  earlier  did  it  rise  on  April  1  than  on 
April  4  ? 

Time  between  events  happening  in  two  different  years. 

Example.  How  many  years,  months,  and  days  were 
between  Aug.  27,  1880,  and  Jan.  22,  1901  ? 

YB.         MO.       DA.          Since  January  is  the  first  month  of 
1901  22     the         r  and  August   is   the   eighth 

1 &&0  &  97 

100U - £i     month  of  the  year,  we  write  1  instead 

25  of  January  and  8  instead  of  August. 
In  finding  the  difference,  a  month  is  taken  as  30  days. 
The  work  is  then  performed  as  in  the  subtraction  of 
compound  quantities. 


EXERCISE   58 

1.  The  battle  of  New  Orleans  was  fought  on  Jan.  8, 
1815.      Find   the   time   from   that   date   to   the   present 
day. 

2.  The  first  telegraph  message  was  sent  by  Professor 
Morse  on  May  24,  1844.     Find  the  time  from  that  date 
to  the  present  day. 

3.  The  Spanish  fleet  under  Cervera  was  destroyed  near 
Santiago  on  July  3,  1898.     Find  the  time  from  that  date 
to  Feb.  1,  1903. 

4.  The  Mecklenburg  Declaration  of  Independence  was 
signed  May  20,  1775.     Find  the  time  from  this  date  to 
the  surrender  of  Cornwallis,  Oct.  19,  1781. 


COMPOUND   QUANTITIES 


99 


5.    The  following  named  men  were  born  and  died  on 
the  dates  specified.     Find  how  long  each  lived. 


John  Milton  .... 
Alexander  Pope      .     . 
William  Shakespeare  . 
Edmund  Burke 
Robert  E.  Lee   .     .     . 
U.  S.  Grant  .... 
Oliver  Goldsmith   . 
Benjamin  Franklin 
Alexander  Hamilton   . 
H.  W.  Longfellow      . 
J.  H.  Newman    .     .     . 
W.  E.  Gladstone    , 


BORN 

Dec.     9,  1608. 

DIED 

Nov.      8,  1674. 

May   21,  1688. 

May     30,  1744. 

April  23,  1564. 

April   23,  1616. 

Jan.    12,  1730. 

July       9,  1797. 

Jan.    19,  1807. 

Oct.     12,  1870. 

April  27,  1822. 

July     23,  1885. 

Nov.  10,  1728. 

April     4,  1774. 

Jan.    17,  1706. 

April   17,  1790. 

Jan.    11,1757. 

July     12,  1804. 

Feb.  27,  1807. 

March  24,  1882. 

Feb.  21,  1801. 

Aug.    11,  1890. 

Dec.     9,  1809. 

May     19,  1898. 

MULTIPLICATION 

Multiply  5  yd.  2  ft.  10  in.  by  7. 

YD.       FT.        IN.          Multiply  10  in.  by  7  and  get  70  in. 
5         2         10      =  5  ft.  10  in.     Write  10  in.  and  carry 

I     5  ft.     7  times  2  ft.  are  14  ft.     14  ft.  and 

10  5  ft.  =  19  ft.  =  6  yd.  1  ft.  Write  1  ft. 
Carry  6  yd.  7  times  5  yd.  are  35  yd.  35  yd.  and  6  yd. 
=  41  yd. 


EXERCISE   59 

Multiply : 

1.  4  yd.  2  ft.  3  in.  by  9. 

2.  6  yd.  1  ft.  9  in.  by  11. 

3.  9  yd.  2  ft.  11  in.  by  8. 

4.  3  bu.  2  pk.  7  qt.  by  7. 


5.  4bu.  Ipk.  6qt.  by  12. 

6.  9  gal.  3  qt.  1  pt.  by  6. 

7.  6  gal.  2  qt.  1  pt.  by  12. 

8.  3  bu.  3  pk.  7  qt.  by  7. 


100  ADVANCED  BOOK  OF  ARITHMETIC 

9.  12  T.  400  Ib.  by  12.  13.  16°  38'  32"  by  15. 

10.  13  T.  387  Ib.  by  9.  14.  64  A.  150  sq.  rd.  by  12. 

11.  17  T.  254  Ib.  by  10.  15.  15  A.  27  sq.  rd.  by  11. 

12.  5°  29'  28"  by  16.  16.  18°  9'  54"  by  14. 

17.  Multiply  |  T.  by  9  and  express  the  result  in  pounds. 

18.  Multiply  ^  mile  by  19  and  give  the  result  in  feet. 

DIVISION 

Divide  97°  10'  50"  by  8. 
n,Q7o     1(V     r0,,  The  eighth  part  of  97°  is  12°,  with 

12°       8'     511"     a  remainder  of  10-     10  10'  =  70'-     The 
4       eighth  part  of  70'  is  8',  with  a  remainder 

of  6'.     6'  50"  =  410".     8  into  410  goes  51 J  times. 

EXERCISE   60 

Divide : 

1.  21  yd.  2  ft.  3  in.  by  9.       4.    34  yd.  2  ft.  8  in  by  6. 

2.  93°  15'  15"  by  7.  5.    77  yd.  2  ft.  4  in.  by  7. 

3.  84°  14' 14"  by  12.  6.    13  bu.  3  pk.  4  qt.  by  6. 

7.  How  often  is  231  cu.  in.  contained  in  1  cu.  ft.  582 
cu.  in.  ? 

8.  How  often  is  7  yd.  1  ft.  contained  in  1  mi.  ? 

9.  A  meter  is  39.37  inches.     How  many  meters  equal 
1  mile  ? 

10.  The  planet  Mercury  revolves  around  the  sun  in  88 
days.     Find  in  degrees,   minutes,  and  seconds  its  daily 
progress. 

11.  Civil  engineers  use  a  chain  100  feet  long.     How 
many  of  these  chains  make  5  miles  ? 


COMPOUND   QUANTITIES  10J 

Example  l.    Reduce  .875  yd.  to  feet  and  inches. 
.875  yd. 

3_ 

2.625  ft.  .625  ft.  =  .625  x  12  in. 

12  =  7.5  in. 

7.500  in. 
.875  yd.  =  2  ft.  7.5  in. 

Example  2.    Reduce  TV  bu.  to  lower  denominations. 
£  bu.  =  J_  Of  4  pk.  =  I  pk.  =  2i  pk. 
1   pk.=   t   ofSqt.   =  fqt.   =  2f  qt, 
|  qt.  =   |  of  2  pt,  =  |  pt.  =  11  pt. 
Hence,  TV  bu.  =  2  pk.  2  qt.  1^  pt.  ' 

Example  3.    Express  -^  A.  in  square  yards. 

7  v  4840 

^  of  1  A.  =  TV  of  4840  sq.  yd.  =  LJ^^^L  sq.  yd. 

12 

sq.  yd.  =  2823^sq.  yd. 


EXERCISE   61 

Reduce : 

1.  %  T.  to  pounds.  6.    . 375  bu.  to  quarts. 

2.  .15°  to  minutes.  '  7.    18.4  mi.  to  feet. 

3.  ^  da.  to  hours  and  minutes.       8.    -f^  mi.  to  yards. 

4.  .2345  T.  to  pounds.  9.    .1875  mi.  to  rods. 

5.  .95  da.  to  hours  and  minutes.   10.    15|°  to  minutes. 

11.  |  of  a  common  year  to  days  and  hours. 

12.  .3125  common  years  to  days,  hours,  and  minutes. 

13.  .45  bu.  to  a  compound  quantity. 

14.  |  gal.  to  a  compound  quantity. 

15.  .85  A.  to  square  rods.  17.   l|  bu.  to  quarts. 

16.  II  A.  to  square  yards.  18.    ^  gal.  to  pints. 


1Q2  ADVANCED  BOOK  OF  ARITHMETIC 

EXPRESSION   OF  ONE   QUANTITY  AS  A   FRACTION  OF 
ANOTHER  QUANTITY 

Example  1.    Express  27  rd.  4  yd.  2  ft.  as  a  fraction  of 
1  mi. 
27  rd.  4  yd.  2  ft. 

52  1  mi  =  5280  ft. 


. 

139  27rd.4yd.  2  ft.  =  of  1  mi. 


1-  yd.  919      .  - 

_  =i0560°flmi- 

4591  ft. 

To  express  one  quantity  as  a  fraction  of  another  quantity, 
reduce  both  to  the  same  denomination,  and  divide  the  first 
quantity  by  the  second. 

Example  2.  Express  2  yd.  2  ft.  8  in.  as  a  decimal  of  a 
mile. 

Step  1.    Express  2  yd.  2  ft.  8  in.  as  a  fraction  of  1  mi. 

Step  2.    Reduce  this  fraction  to  a  decimal. 

Reduce:  EXERCISE  62 

1.  4400  ft.  to  the  decimal  of  1  mi. 

2.  293  yd.  1  ft.  to  the  decimal  of  1  mi. 

3.  117  yd.  1  ft.  to  the  decimal  of  1  mi. 

4.  1  qt.  1  pt.  to  the  decimal  of  a  gal. 

5.  1  pk.  6  qt.  to  the  decimal  of  a  bu. 

6.  2  pk.  2  qt.  to  the  decimal  of  a  bu. 

7.  3  pk.  1  qt.  1  pt.  to  the  decimal  of  a  bu. 

8.  1^  in.  to  the  decimal  of  1  ft. 

9.  2.34  in.  to  the  decimal  of  1  ft. 

10.  3°  15'  to  the  fraction  of  a  right  angle. 

11.  1  da.  18  hr.  to  the  decimal  of  1  wk. 


MEASUREMENTS 


103 


MEASUREMENTS 
EXERCISE   63 

1.  Find  the  number  of  acres  in  the  area  of  a  rectangle 
whose  dimensions  are  360  ft.  and  121  ft. 

2.  Find  the  area  of  a  rectangle  1331  ft.  by  720  ft. 

3.  Find,   in  acres,  the   area  of  a  rectangular  plot  of 
ground  201  yd.  by  10  rd.  5  yd. 


6 

5 

4 

3 

2 

1 

7 

8 

9 

10 

11 

12 

18 

17 

16 

15 

H 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

FIG.  1. 


4.  A  railway  company  acquires  the  right  of  way 
through  a  territory  154  mi.  long,  and  fences  in  a  strip 
80  ft.  wide.  How  many  acres  does  it  thus  inclose,  and 
how  much  does  it  pay  for  the  land  at  $  25  an  acre  ? 


104 


ADVANCED  BOOK  OF  ARITHMETIC 


5.  Find  the  area  of  a  park  396  ft.  by  396  ft.     Give 
your  answer  in  acres. 

6.  How  many  acres  are  in  a  rectangular  farm  1.5  mi. 
long  by  1|  mi.  wide  ?     Find  the  value  of  the  farm  at  $49 
an  acre. 

A  township  is  a  tract  of  land  6  mi.  square,  and  it  is 
divided  into  36  sections  each  1  mi.  square.  The  sections 
are  numbered  as  in  Fig.  1. 


NJ/20f 

N.W.  y4 

S.  W.  t/4  of 

S.  E.  1/4  of 

N.  W.i/4 

N.  W.1/4 

s.  w.  y4 

FIG.  2. 

A  section  is  subdivided  as  indicated  in  Fig.  2.  There 
are  two  divisions,  E.  |  and  W.  J.  The  E.  J  is  divided 
into  two  equal  squares  called  N.E.  \  and  S.E.  \.  The 
W.  J  is  divided  into  two  equal  squares  called  N.W.  ^ 
and  S.W.  \.  These  are  again  subdivided  as  shown. 


MEASUREMENTS 


105 


EXERCISE  64 

1.  Draw   a   figure  and  locate  S.W.  ^  of  S.E.  J;    N. 
W.  \  of   S.E.  |  ;     N.E.   1    of   S.W.  J;     S.E.   J   of  S. 
W.I 

2.  Locate  N.E.  J  of  the  N.E.  \  ;     S.W.  £  of  N.E.  -|  ; 
How  many  acres  are  in  N.W.  \  of  S.W.  |  ?     How  many 
acres  are  in  N.W.  J  ?  in  S.E.  *  ?  in  S.  J  of  S.W.  £? 

3.  A  man  buys  |  of  the  S.  J  of  N.W.  ^  and  also  |  of  the 
N.W.  l  of  S.W.  \  at  the  rate  of  $.20  an  acre.     Find  the 
cost  of  his  purchase. 

VOLUMES  OF  RECTANGULAR   SOLIDS 

The  volume  of  a  rectangular  solid  is  obtained  by  taking 
the  product  of  its  three  dimensions  expressed  in  units  of 
the  same  denomination. 

EXERCISE  65 

1.  Find  the  volume  of  a  box  12 
ft.  by  7  ft.  by  6  ft. 

2.  Find  the  cubical  contents  of 
a  room  18  ft.  by  12  ft.  and  9  ft. 
high. 

3.  Find    the    number   of    cubic 
feet  in  a  room  32  ft.  by  24  ft.  and  12  ft.  high. 

4.  How  many  cubic  yards  of  earth  must  be  removed 
for  the  foundation  of  a  house  75  ft.  by  54  ft.,  if  the  earth 
to  the  depth  of  21  ft.  is  removed  ? 

5.  A  cistern,  in  the  shape  of  a  rectangular  solid,  is  22 
ft.  by  14  ft.  and    6    ft.    deep.     How   many   gallons  of 
water  does  it  contain? 


106  ADVANCED  BOOK  OF  ARITHMETIC 

6.  A  bin  is  8  ft.  by  3  ft.  and  6  ft.  high.     How  many 
bushels  does  it  hold  ? 

7.  In  order  to  build  a  concrete  wall,  earth  is  removed 
to  the  depth  of  6  ft.     If  the  wall  is  210  ft.  long  and  12  ft. 
wide,  how  many  cubic  yards  of  earth  must  be  removed  ? 

8.  How  many  cubic  yards  of  gravel  are  required  to 
fill,  to  the  depth  of  6  in.,  a  street  1  mi.  long  and  36  ft. 
wide  ? 

9.  How  many  cubical  boxes  2  ft.  each  way  would  a 
storeroom  18  ft.  by  12  ft.  and  10  ft.  high  hold  ? 

10.  The  Sault  Ste.  Marie  Canal  is  1.6  mi.  long,  160  ft. 
wide,  and  25  ft.  deep.     Express  in  cubic  yards  the  volume 
of  water  required  to  fill  it. 

11.  A  block  of  marble  is  4  ft.  by  3  ft.  and  24  ft.  long. 
How  many  tons  does  it  weigh,  if  a  cubic  foot  of  marble 
weighs  170  Ib.  ? 

12.  How  many  pounds  does  a  cedar  beam  14  in.  by  10 
in.  and  40  ft.  long  weigh,  if  a  cubic  foot  of  cedar  wood 
weighs  38.1  Ib.  ? 

13.  A  cubic  foot  of  clay  weighs  75  Ib.     Find,  in  tons, 
the  weight  of  a  clay  bank  10  ft.  by  4  ft.  and  80  ft.  long. 

14.  A  box  9  in.  by  8  in.  and  6  in.  deep  is  filled  with 
mercury.     Find  its  weight  in  pounds  if  a  cubic  foot  of 
mercury  weighs  13,570  oz. 

15.  How  many  3-in.  cubes  are  required  to  fill  a  cubical 
box  each  of  whose  edges  is  1  yd.  ? 

16.  A  pile  of  4-ft.  wood  8  ft.  long  and  4  ft.  high  con- 
tains a  cord.     How  many  cords  of  wood  are  in  a  pile  of 
4-ft.  wood  120  ft.  long  and  12  ft.  high  ? 

17.  Find  the  weight  of  the  water  covering  an  acre  to 
the  depth  of  4  inches.     1  cu.  ft.  of  water  weighs  1000  oz. 


MEASUREMENTS 


107 


FIG.  1. 


FIG.  2. 


A  triangle  is  a  portion  of   a  plane  bounded  by  three 
straight  lines.     ABC  is  a  triangle. 

Two  straight  lines  are  parallel  if 
they  can  never  meet  no  matter  how 
far  they  may  be  produced. 

A  quadrilateral  is  a  portion  of 
a  plane  bounded  by  four  straight 
lines.  Figure  2  represents  a 
quadrilateral. 

A  quadrilateral  having  its 
opposite  sides  parallel  is  called 
a  parallelogram.  Figure  3  is  a 
parallelogram. 

The  sides  AB,  CD  are  paral- 
lel. Also  the  sides  AD,  BO 
are  parallel.  t 

Consider  next  the 
parallelogram  A  B  CD  M_  D 
(Fig.  4)  and  the  rec- 
tangle ABKM.  By 
actual  trial  the  tri- 
angle BKC  is  equal 
to  the  triangle  AMD. 
Take  the  triangle 
AMD  from  the  figure  ABCM,  the  parallelogram  remains. 
If  the  triangle  BKQ  is  taken  from  the  figure  ABCM,  the 
rectangle  remains.  Hence,  the  rectangle  equals  the  paral- 
lelogram in  area.  But  the  area  of  the  rectangle  is  obtained 
by  taking  the  product  of  its  two  dimensions,  i.e.  AB  and 
BK.  Therefore,  the  area  of  the  parallelogram  is  equal  to 
AB  x  BK.  AB  is  called  the  base  of  the  parallelogram ; 
BK,  i.e.  the  distance  between  the  parallel  sides,  is  called 
the  altitude,  or  height,  of  the  parallelogram. 


FIG.  3. 


K 


FIG.  4. 


108 


ADVANCED   BOOK  OF  ARITHMETIC 


FIG.  5. 


Consequently,  the  area  of  a  parallelogram  equals  the 
product  of  its  base  by  its  altitude. 

A  quadrilateral  having  two  sides  parallel  is  called  a 
trapezoid.  ABCD  (Fig.  5)  is  a  trapezoid,  having  AB 

parallel  to  DC.  AB 
and  CD  are  respec- 
tively the  lower  and 
upper  bases  of  the 
trapezoid. 

Take  a  piece  of 
paper  and  make  a 
trapezoid.  Make  an- 
other trapezoid  just 
equal  to  it. 

^  Place  them  as  shown 

in  Fig.  6.     You  then 

have  a  parallelogram  KBCL.     Its  area  is  equal  to 
-f-  AB)  x  height  of  the  parallelogram,  i.e.  equal  to 
AB)  x  height  of  the  parallelogram. 
The  trapezoid  is  -|-  of  KBCL. 

.-.  the  area  of  the  trapezoid  ABCD  equals  |  of  (DC  + 
AB)  x  the  height  of  the  trapezoid. 

Consequently,  the  area  of  a  trapezoid  equals  one  half 
the  sum  of  its  parallel  sides  multiplied  by  the  distance 
between  them. 

If  one  of  the  bases,  e.g.  DC,  of  a  trapezoid  were  to 
become  smaller  and  smaller,  the  fig- 
ure would  ultimately  be  a  triangle. 
Hence,  the  area  of  a  triangle  equals 
one  half  of  its  base  multiplied  by  its 
height. 

This  may  also  be  seen  readily  from     - 
Fig.  7.  FIG.  7. 


MEASUREMENTS  109 

EXERCISE  66 
Find  the  areas  of  the  following  parallelograms : 

1.  Base  10'  6",  altitude  6'  4".* 

2.  Base  17'  3",  altitude  9'  8". 

3.  Base  12'  6",  altitude  8'  3". 

4.  Base  15'  5",  altitude  9'  4". 

5.  Base  36'  9",  altitude  8'  4". 

6.  Base  40'  3",  altitude  7'  6". 

7.  Base  27'  9",  altitude  8'  7". 

8.  Base  28'  4",  altitude  6'  3". 

Find  the  areas  of  the  following  triangles : 

9.  Base  50'  3",  altitude  23'  9". 

10.  Base  60'  4",  altitude  42'  8". 

11.  Base  75'  6",  altitude  35'  8". 

12.  Base  48'  4",  altitude  29'  4". 

13.  Base  56'  9",  altitude  27'  4". 

14.  Base  82'  6",  altitude  64'  2". 

15.  Find  the  area  of  the  upper  surface  of  a  board  in  the 
form  of  a  trapezoid  whose  parallel  sides  are  12'  6"  and  5' 
6"  and  height  8'  6". 

16.  Find  the  area  of  a  trapezoid  whose  parallel  sides 
are  18'  4"  by  12'  8"  and  24'  apart. 

17.  Find  the  area  of  a  field  whose  parallel  sides  are 
20  rd.  and  36  rd.  and  width  18  rd. 

18.  Find  the  area  of  a  lot  whose  shape  is  a  trapezoid 
having  for  parallel  sides  60  ft.  and  40  ft.  and  length  84  ft. 

*  6  ft.  4  in.,  a  notation  used  by  engineers,  architects,  and  mechanics. 


110  ADVANCED  BOOK  OF  ARITHMETIC 

19.  Find  the  area  of  a  right  triangle  whose  base  is  26 
ft.  and  altitude   19   ft.     (A  right  triangle  is  a  triangle 
having  a  right  angle.) 

20.  Find  the  area  of  a  right  triangle  whose  base  is  96  ft. 
6  in.  and  altitude  84  ft. 

21.  Find  the  area  of  a  right  triangle  having  for  base 
72  ft.  6  in.  and  altitude  63  ft.  9  in. 


BOARD  MEASURE 

Lumber  is  measured  by  the  board  foot.  A  board  foot  is 
a  rectangular  solid  12  in.  by  12  in.  and  1  in.  in  height. 

If  a  board  is  1  in.  thick,  the  number  of  board  feet  in  it 
is  measured  by  the  number  of  square  feet  in  its  upper  or 
lower  surface.  Thus,  a  board  8  ft.  long,  8  in.  wide,  and 
1  in.  thick  contains  8  x  f  board  feet.  If  a  board  is  more 
than  1  in.  thick,  the  number  of  board  feet  in  it  is  measured 
by  the  area,  in  square  feet,  of  its  upper  or  lower  surface 
multiplied  by  the  number  denoting  the  thickness  in  inches. 
Thus,  a  board  9  ft.  by  14  in.  and  2|-  in.  thick  contains 
9  x  if  x  21  board  feet. 

The  number  of  board  feet  in  a  board  less  than  1  in.  in 
thickness  is  measured  by  the  number  of  square  feet  in  its 
upper  or  lower  surface.  Thus,  a  board  10  ft.  by  15  in. 
and  |-  in.  thick  contains  10  x  ^f  board  feet. 

EXERCISE   67 

1.  How  many  board  feet  are  in  a  board  8  ft.  by  1J  ft. 
and  1  in.  thick  ? 

2.  How  many  board  feet  are  in  a  board  9  ft.  by  16  in. 
and  |  in.  thick?     Find  the  cost  of  the  board  at  3|^  per 
board  foot. 


MEASUREMENTS  111 

3.  How  many  board  feet  are  in  a  plank  24  ft.  by  15 
in.  and  2  in.  thick  ?     In  a  board  20  ft.  by  12  in.  and  3  in. 
thick  ? 

4.  How  many  board  feet  are  in  a  railroad  tie  24  ft. 
by  8  in.  and  6  in.  thick? 

5.  Find  the  number  of  board  feet  in  each  of  the  follow- 
ing pieces  of  lumber: 

(a)    18  ft.  by  16  in.  and  1J  in.  thick. 
(5)    12  ft.  by  8  in.  and  6  in.  thick. 
0?)    24  ft.  by  9  in.  and  3  in.  thick, 
(d)    16  ft.  by  8  in.  and  4  in  thick. 
00    18  ft.  by  12  in.  and  3  in.  thick. 
(/)    21  ft.  by  16  in.  and  3  in.  thick. 
(jf)    30  ft.  by  14  in.  and  21  in.  thick. 
(i)    28  ft.  by  15  in.  and  3J  in.  thick. 

6.  Find  the  cost  of  480  boards,  each  11  in.   by  10  in. 
and  16  ft.  long,  @  127.50   per   M.     ("Per  M"  means 
"  by  the  1000  "  board  feet.) 

7.  Find  the  cost  of  840  boards,  each  If  in.  by  10  in. 
and  12  ft.  long   @  1 25  per  M. 

8.  How  many  board  feet  are  in  a  board  24  ft.  long,  12 
in.  wide  at  one  end  and  16  in.  wide  at  the  other  end,  and 
2 1  in.  thick  ? 

9.  How  many  board  feet  are  in  a  board  18  in.  wide  at 
one  end,  12  in.  wide  at  the  other  end,  2J  in.  thick,  and  28 
ft.  long  ? 

10.  How  many  board  feet   are   in  a  cubical   block  of 
wood,  each  of  whose  dimensions  is  2  ft.? 

11.  How  many  board  feet  are  in  a  beam  9  in.  by  9  in. 
and  54  ft.  long  ? 

12.  How  many  board  feet  are  in  16  railroad  ties  8"  by 
6"  and  8'  6"  long  ? 


112  ADVANCED  BOOK  OF  ARITHMETIC 

13.  How  many  cubic  feet  are  equivalent  to  120  board 
feet? 

14.  How  many  board  feet  are  in  a  beam  11  in.  by  12  in. 
and  36  ft.  long  ? 

15.  A  piece  of  lumber  contains  2980  cu.  in.     Express 
this  in  board  feet,  understanding  that  the  piece  is  more 
than  1  in.  in  thickness. 

16.  Find  the  number  of  board  feet  in  a  beam  12  in.  by 
6  in.  at  one  end,  9  in.  by  6  in.  at  the  other  end,  and  27  ft. 
long. 

MASONRY  AND  BRICKLAYING 

Stone  work  is  estimated  by  the  perch,  also  by  the  cubic 
foot. 

A  perch  of  masonry  is  24f  cu,  ft.  It  is  a  wall  1  rd. 
long,  1|-  ft.  wide,  and  1  ft.  high.  Perch  is,  in  Great 
Britain,  another  name  for  rod.  Of  the  24|  cu.  ft.  in  a 
perch,  22  cu.  ft.  are  allowed  for  stone  and  2|  cu.  ft.  are 
allowed  for  mortar,  i.e.  eight  ninths  for  stone  and  one 
ninth  for  mortar. 

In  estimating  the  number  of  perches  of  masonry  in  the 
walls  of  a  building,  the  outside  dimensions  of  the  walls 
are  taken.  This  method  of  reckoning  counts  the  corners 
twice.  In  estimating  the  amount  of  material,  the  com- 
putation should  be  exact,  inside  and  outside  dimensions 
being  reckoned. 

Twenty-two  common  bricks,  i.e.  bricks  8  in.  by  4  in.  by 
2  in.,  are  reckoned  as  a  cubic  foot. 

EXERCISE   68 

l*  How  many  perches  of  masonry  are  in  a  wall  84  ft. 
long,  16  ft.  high,  and  1J  ft.  thick  ?  How  many  common 
bricks  are  required  to  build  this  wall  ? 


MEASUREMENTS 


113 


2.  A  cellar  is  36  ft.  long,  18  ft.  wide,  and  8  ft.  deep. 
How  many  cubic  feet  of  masonry  are  in  its  walls  if  they 
are  2  ft.  thick  ?     What  is  the  actual  number  of  cubic  feet 
in  the  walls  ? 

3.  How  many  bricks  are  necessary  to  build  a  wall  25 
ft.  long,  12  ft.  high,  and  24  in.  in  thickness  ?     What  is 
the  cost  of  the  bricks  at  $9  per  M  ? 

4.  How  many  cubic  feet  of  mortar  are  required  to  build 
a  wall  600  ft.  long,  10  ft.  high,  and  18  in.  in  thickness  ? 
How  many  common  bricks  are  required  to  build  this  wall  ? 

5.  At  $9.50  per  M,  what  is  the  cost  of  the  bricks  re- 
quired to  build  a  house  40  ft.  by  36  ft.  and  40  ft.  high  to 
the  eaves,  the  highest  point  of  the  gable  end  being  52  ft. 
above  the  ground,  and  the  walls  1^  ft.  thick  ?     Allow  270 
cu.  ft.  for  openings. 

CARPETING 

1.    How  many  yards  of  carpet  are  needed  for  a  room 
18  ft.  by  12  ft.,  if  the  carpet  is  27  in.  wide  ? 

18ft. 


12ft. 


SOLUTION.     (1)    Let  the  carpet  be   placed  crosswise. 
Number  of  strips  =  18  ft.  -^  27  in.  =  18  ft.  -*-  2£   ft.  =  8. 
.• .  Number  of  yards  =  12  x  8  -r-  3  =  32.     Ans.  32  yd. 


114  ADVANCED   BOOK  OF  ARITHMETIC 

(2)  Let  the  carpet  be  placed  lengthwise. 

g  ft  Number  of  strips  : 


12ft.  -5-  21  ft.  =5^. 

Here    6    strips    are 
needed.     The  fraction 
12  ft.    of  a  strip  may  be  either 
cut  off  or  turned  under. 
.-.  Number  of  yards 
=  6  yd.  x  6  =  36  yd. 
Ans.  36  yd. 

To   find  the  number   of  yards   of  carpet  required  to  carpet 
a  room: 

1.  Draw  a  diagram  of  the  room. 

2.  Find  the  number  of  strips. 

3.  Multiply  the  number  of  strips  by  the   number  of 
yards  in  one  strip. 

EXERCISE   69 

1.  How  many  yards  of  carpet  are  required  to  cover  a 
room  18  ft.  by  16  ft.  with  carpet  30  in.  wide,  if  the  strips 
are  laid  crosswise  ? 

2.  How  many  yards  of  carpet  f  of  a  yard  wide   are 
needed  to  carpet  a  room  24  ft.  by  18  ft.,  if  the  carpet  runs 
lengthwise  ? 

3.  How  many  yards  of  matting  36  in.  wide  are  required 
to  cover  a  room  16  ft.  by  12  ft.,  the  matting  being  laid 
crosswise  ? 

4.  How  many  yards  of  carpet  30  in.  wide  are  needed 
to  carpet  a  room  24  ft.  long,  20  ft.  wide,  if  the  carpet  is 
laid  crosswise  ? 

5.  Find  the  cost  of  carpeting  a  room  16  ft.  by  14  ft. 
with  carpet  27  in.  wide  at  90  ^  per  yard,  the  strips  being 
laid  lengthwise. 


MISCELLANEOUS  EXERCISES  115 

MISCELLANEOUS  EXERCISES 
EXERCISE  70 

1.  How  long  will  it  take  a  person  to  earn  $44.40,  if 
he  earns  11.85  a  day  ? 

2.  If  |  of  a  quantity  of  coal  is  71 J  tons,  how  many 
tons  of  coal  are  there  in  all  ? 

3.  A    train    runs    at   the  rate   of  33^  mi.    per  hour. 
How  many  hours  will  it  take  the  train  to  run  350  mi.? 

4.  A  man  being  asked  his  age  replied,  "  Three  elev- 
enths of  my  age  is  13 J  years."     Find  the  man's  age. 

5.  A's  share  is  f  of  the  joint  capital,  or  $2540.     What 
is  the  joint  capital  ? 

6.  A  dealer  sells  a  piano  for  $240,  thereby  gaining  ^  of 
the  cost  of  the  piano.     What  did  the  piano  cost  ? 

7.  If  a  dealer  sold  a  piano  for  $240  and  lost  thereby  -J 
of  the  cost  price,  what  did  the  piano  cost  ? 

8.  If  the  dividend  is  18f  and  the  quotient  is  10J,  what 
is  the  divisor  ? 

9.  The  sum  of  two  numbers  is  25,  and  one  of  the  num- 
bers is  1^-  times  the  other  number.     Find  the  numbers. 

10.  A  farmer  sells  |  of  his  cattle  to  one  jobber  and  -|  to 
another  jobber.     If  he  keeps  the   remainder,  what  part . 
does  he  keep  ?     Suppose  the  number  remaining  is  27,  how 
many  head  of  cattle  had  the  farmer  originally  ? 

11.  A  dealer  sells  |  of  his  coal,  and  then  ^  of  it,  and  has 
12|-  tons  left.     How  many  tons  had  he  originally  ? 

12.  If  the  ^  part  of  a  number  and  the  1  part  of  the  same 
number  together  make  60J,  what  is  the  number  ? 

13.  If  .85  of  A's  money  is  $  289,  how  much  has  he  ? 


116  ADVANCED  BOOK  OF  ARITHMETIC 

14.  By   gaining  .15  of    his   outlay  a    man's    property 
amounts  to  $7245.     What  was  his  outlay? 

15.  Five  sevenths  of  a  farm  is  sold  for  $1385.     What 
is  the  value  of  the  farm  ? 

16.  How  many  times  will  a  wheel  12  ft.  4^  in.  in  cir- 
cumference revolve  in  going  3|  mi.  ? 

17.  How  long  will  it  take  a  person  to  go  10|  mi.  at 
the  rate  of  6^  mi.  per  hour  ? 

18.  A  train    travels  at  the  rate  of  52^  mi.  in  If  hr. 
Find  its  rate  per  hour.     How  many  hours  and  minutes 
will  it  take  this  train  to  travel  208  mi.  ? 

19.  From  the  ground  to  the  first  floor  of  a  house  is  -^ 
of  the  height  of  the  house;  if  the  first  floor  is  10  ft.  above 
the  ground,  how  high  is  the  house  ? 

20.  What  part  of  4f  is  J-  of  14-|  ? 

21.  If  .625  of  a  gallon  of  maple  sirup  cost  $1.25,  what 
will  1  gal.  of  maple  sirup  cost  ? 

22.  When  3  oranges  sell  for  5^,  how  many  oranges  can 
I  buy  for  80^? 

23.  If  8^  yd.  of  calico  cost  66^,  how  many  yards  can 
I  buy  for  $3.30? 

24.  A  merchant,  by  selling  tea  at  69^  per  pound,  gains 
2^-  of  the  cost  of  the  tea.     Find  the  cost  per  pound  of  the 
tea. 

25.  By  selling  cloth  at  78^  per  yard  a  clothier  loses  ^ 
of  the  cost  of  the  cloth.     Find  the  cost  price  per  yard  of 
the  cloth. 

26.  The  rent  of  a  house  for  17  mo.  is  $225.     Find  the 
rent  of  this  house  for  12  mo. 

27.  Lead  pencils  cost   3^  each.      At  this  price,  how 
many  can  be  bought  for  $1.40? 


MISCELLANEOUS  EXERCISES  117 

28.  A  laborer  gets  4  ^  per  cubit  foot  for  digging  a  cel- 
lar.    At  this  rate  ho^  much  will  he  get  if  the  cellar  is 
15  ft.  by  12  ft.  and  6  ft.  deep? 

29.  If  3  qt.  of  oil  cost  13  J  ^,  at  this  rate  how  much  will 
1  gal.  2  qt.  cost  ? 

30.  If  1  Ib.  8  oz.  of  cheese  cost  42  ^,  find  the  cost  of  4  Ib. 

31.  How  many  tiles  6  in.  square  will  be  needed  to  pave 
a  hearth  5  ft.  by  2  ft.  ? 

32.  Express  ^  +  -|  -f  ^  as  a  decimal. 

33.  Reduce  5.875  hr.  to  seconds. 

34.  Find  the  cost  of  a  car  load  of  coal  weighing  35,880 
Ib.  at  $5.50  per  ton. 

35.  A   man   having  a  salary  of   $1500  a  year  spends 
25%  of  it  for  board,  10%  for  clothing,  and  15%  for  other 
things.     How  much  money  does  he  spend  ? 

36.  Reduce  2876  in.  to  yards. 

37.  Find  the  value  of  32  tubs  of  butter  each  weighing 
56  Ib.  at  37|  #  per  pound. 

38.  The  wages   of   a  motorman   on   the    Metropolitan 
West  Side  Elevated  Railway,  Chicago,  are  30^  ^  per  hour. 
How  much  does  he  earn  in  ten  weeks,  working  8  hr.   a 
day  ? 

39.  How  many  feet  are  in  87-|  %  of  1  mile  ? 

40.  How  many  square  yards  are  in  62|-%  of  1  A.  ? 

41.  A  farmer  having  320  bushels  of  apples  sells  75%  of 
them.     How  many  bushels  does  he  sell  ? 

42.  How  much  will  a  man  earn  in  10  weeks  at  $1.50 
per  day,  Sundays  excepted  ? 

43.  A   cubit  foot  of  air  weighs  .08073  Ib.     Find  the 
weight  of  the  air  in  a  room  15  by  16  ft.  and  9  ft.  high  ? 

44.  How  many  grains  are  in  83^%  of  1  Ib.  Troy  ?  , 


118  ADVANCED   BOOK   OF  ARITHMETIC 

45.  A  railroad  train  runs  1  mi.  in  1  min.  15  sec.    Find 
its  rate  per  hour. 

46.  Find  the  number  of  acres  in  a  field  90  rd.  by  40  rd. 

47.  Find  the  cost  of  18,750  bd.  ft.  of  lumber  at  $30  per  M. 

48.  Find  the  commission  on  sales  amounting  to  $  4750 
at  2%. 

49.  A  man  borrows  money  on  April  1,    and  agrees  to 
pay  it  in  90  days.     On  what  date  should  he  pay  it  ? 

50.  How    many  square  inches  are  in  the  surface  of  an 
18-in.  cube? 

51.  Find  in  square  rods  15%  of  1  A. 

52.  Out  of  a  class  of  64  pupils  12J%  failed  to  be  pro- 
moted.    How  many  were  promoted  ? 

53.  A  lot  is  42  ft.  by  120  ft.     Find  the  cost  of  fencing 
it  at  85  ^  per  yard. 

-  54.    Find  the  cost  of  making  a  concrete  sidewalk  63  by 
12  ft.  at  $  1.75  per  square  yard. 

55.  Express  in  minutes  .075  of  1  day,  16  hours. 

56.  A    railroad  train    runs  72  ft.  in  one  second.     At 
this  rate  how  far  will  it  run  in  one  hour  ? 

57.  Find  the  cost  of  plastering  the  walls  of  a  room  18 
ft.  by  20  ft.  and  10  ft.  high  at  25^  per  sq.  yd. 

58.  A  watch  loses  1  min.  and  5  sec.  in  3  da.     At  this 
rate  how  much  will  it  lose  during  the  month  of  April  ? 

59.  How  many  pounds  avoirdupois  are  in  420  Ib.troy  ? 

60.  A  tank  is  8  ft.  by  6  ft.  and  7  ft.  deep.     How  many 
gallons  does  it  hold?     (1  cu.  ft.  =  7J  gal.  nearly.) 

61.  Find  the  cost  of  24  yd.  of  cloth  at  f  1.87J-  per  yd. 

62.  How  many  cords  of  wood  are  in  a  pile  50  ft.  long, 
12  ft.  wide,  and  8  ft.  high  ? 


MISCELLANEOUS   EXERCISES  119 

63.  When  coal  sells  for  $  7.00  per  ton,  what  is  the  cost 
of  a  sack  of  coal  weighing  200  Ib.  ? 

64.  A  bin  is  18  ft.  long,  6  ft.  wide,  and  4  ft.  deep.     How 
many  bushels  will  it  contain?    (1  bu.  =  1-J-  cu.  ft.) 

65.  Change  to  decimals  J£,  ||,  ^f 7,  f . 

66.  Change  to  fractions  in  lowest  terms  .875,  .00525, 
.66|. 

67.  Express  in  simplest  form 

7  x  3.04  x  10,000  -  125,000.7. 

68.  A  man  rents  a  house  for  $  550  a  year.     His  rent  the 
previous  year  was  9%  less.     What  was  his  rent  last  year? 

69.  A  grocer  having  on  hand  15  gal.  2  qt.  1  pt.  of  oil 
buys  20  gal.  2  qt.,  and  sells  30  gal.  2  qt.  1  pt.     How 
much  has  he  left  ? 

70.  Find  %\%  of  1  cu-  ft.     Give  result  in  cubic  inches. 

71.  The  following  is  the  lowest  bid  in  detail  for  improv- 
ing Lick  Run  Pike,   Cincinnati,    Ohio,    submitted    July 
19,  1907 : 

20,000  cu.  yd.  embankment  @  $  .018J 
500  cu.  yd.  excavation  @  $1.00 
1530  cu.  yd.  stone  @  $2.25 
265  cu.  yd.    screening  @  $2.25 
100  cu.  yd.  cement  @  $8.00 
400  ft.  12"  pipe  @  $  1.00 
10  ft.  12"  slant  @$. 80 
30  ft.  4"  box  culvert  concrete  @  $  6.00 
60  ft.  5"  box  culvert  concrete  @  $7.00 
2350  ft.  7"  box  culvert  concrete  @  $16.00 
650  ft.  8"  box  culvert  concrete  @  $18.00 
18,400  sq.  yd.  rolling  @  3^ 
6  manholes®  $40.00 
Find  the  amount  of  the  bid. 


120  ADVANCED  BOOK  OF  ARITHMETIC 

REVIEW   OF   FRACTIONS 

EXERCISE   71 
Add: 

1>      6'  A'  ITP    lV  7< 

2.    2f,324,4ix.  8. 

3-    i>  iV'  A'  iV  9-    I'  I3!'  TO' 

4.   If,  iWi'liV  10.   ^,3T 


6.  2i|,  2||,  42V  12.  2TV,  3if,  21  if  . 

Find  the  difference  between  : 

13.  4T\  and  If.  22.  J|  and  if 

14.  l^andTW  23.  Handlf 

15.  l^andTV  24.  7f  and  311 

16.  -3*5-  and  T^.  25.  8  1  and  711. 

17.  1  and  -^L.  26.  9T52  and  6if  . 

18.  1  and  ^g-.  27.  5T7g  and  2||. 

19.  -^  and  f  .  28.  6T^  and 


20.  fandf  29. 

21.  11  and  if  30.    12^-  and  8|f. 

31.  What  number  must  be  added  to  3f  to  give  5f  ? 

32.  A  man's  capital  amounts  to  $  1727J.     By  how  much 
must  he  increase  his  capital  so  that  it  may  amount  to 
$3000? 

33.  What  number  must  be  taken  from  30|  to  leave  14|? 

34.  By  how  much  does  84^  exceed  17|? 

35.  Of  the  weight  of  the  earth's  atmosphere   y%3oV  ^s 
oxygen.     What  fraction    of    the    weight    of    the    earth's 
atmosphere  do  its  other  constituents  aggregate  ? 


REVIEW  OF  FRACTIONS 


121 


Simplify : 

1.  foffof2f 

2.  f  ofT^of4f 

3.  I  of  1|  of  £$. 

b£4f 


EXERCISE  72 


TT 


xT3Tx4|. 


4. 
5. 
6. 
7. 
8. 
9. 

10.    2|x3^x 

11  ^1    V   ^1    V 

-I.J..         t-JS"  /N   '•^9'    ^ 

12.  2|x2|x 

13.  74-xiSx 


14.  9^ 

15.  1\ 

16.  (i 


18. 
19. 

of  4|  of  If     20. 

v  £\1   v  1  1  0  O1 

S\    ^^    /N    J-"5-5".  <iJ.. 

of  1|  of 


xlfxlfxl2V 


22.    2ix(lf  +  2i  +  l£) 

23. 
24. 
25. 
26. 


EXERCISE   73 


Divide  : 
l.    25fbylf. 
2.    3lby2|. 
3.    IQi  by  Iff. 
4.    7T5g  by  2921 

9-    tobyT5iofll. 
10.    2l|  by  1||  of  If. 
ll.    If  by  f  of  If 
12.    Sy3^  by  Iy8y  times 

5.    521  by  1311        13.    (T9__l)by(|  +  l). 


6. 
7. 


of6f. 


by  t  of 


14. 
15. 


BiV-^bySl 
8.    l  by  f  of  If       16.    lllby(l|  +  ll). 

17.  If  the  dividend  is  6J  and  the  quotient  1|,  find  the 
divisor. 

18.  What  must  10|  be  divided  by  to  give  as  quotient  2|? 


122  ADVANCED  BOOK  OF  ARITHMETIC 

PERCENTAGE 

The  result  of  taking  a  rate  per  cent  of  a  quantity  is 
percentage.  A  percentage  of  a  number  is  simply  a  fraction 
of  the  number.  The  central  fact  in  percentage  is  that 
100  %  is  equivalent  to  1,  or  1  %  is  equivalent  to  yj^-. 

What  per  cent  is  -||  equivalent  to  ? 

^ofl  =  |f  of  100%  =  95%. 

Example  1.    Find  2f  %  of  1789. 
17.89=1%  of  1789. 


_    _ 

35.78  2  times  17.89. 

8.945  i  of  17.89. 

4.4725  of  17.89. 


49.1975  =  2|  times  17.  89. 
Example  2.    Find  3.6  %  of  2992. 

2992  x  M  =  2992  x  .036  =  107.712,  or 

1  %  of  2992  =  29.92  ;  29.92  x  3.6  =  107.712. 

EXERCISE  74 
Find: 

1.  7%  of  184.  10.  10%  of  7850. 

2.  9%  of  275.  11.  11%  of  983. 

3.  6%  of  213.  12.  12%  of  2570. 

4.  8  %  of  534.  is.  12|  %  of  928. 

5.  9%  of  3280.  14.  231%  Of  5220. 

6.  8%  of  3297.  is.  16f  %  of  733. 

7.  4%  of  615.  16.  37J%of828. 

8.  7  %  of  2630.  17.  62|  %  of  9200. 

9.  9%  of  4280,  18.  60%  of  2855. 


PERCENTAGE  123 

19.  4-J%  of  2280.  22.   5f  %  of  10,656. 

20.  3f  %  of  3066.  23.    |-%  of  1690. 

21.  6J%  of  1820.  24.    1%  of  8124. 

25.  Express  the   following   fractions  as  per  cents: 

i'  i>  i>  i>  iV»  A'  iV- 

26.  Express  each  of  the   following   decimals  as  a  per 
cent: 

.04,  .08,  .121,  .0165,  .002,  .006,  .0024. 

27.  The  composition  of  a  piece  of  coal  taken  from  the 
Texas  and  Pacific  Coal  Company's  mine  is  given  as  fol- 
lows:   moisture,    5.46%  ;    combustible    matter,   35.66%  ; 
fixed  carbon,  49.17%  ;  ash,  9.71%.      Find  the  amount  of 
each  constituent  in  2000  Ib.  of  coal.     Check  your  answer. 

28.  The  analysis  of  a  specimen  of  lignite  is  given  as 
follows:  moisture,  29.07%;   combustible  matter,  28.96%  ; 
fixed  carbon,  24.47%  ;  ash,  17.50%.     Find  the  amount  of 
each  constituent  in  a  ton  of  lignite.     Check. 

29.  Distilled  water  is  composed  of  two  gases,  H|%  by 
weight  being  hydrogen,  and  88f  %  by  weight  being  oxy- 
gen.    Find  the  weight  of  each  gas  that  can  be  obtained 
from  10  Ib.  of  water. 

30.  A  bookkeeper  receives  a  salary  of  $  1800  per  annum. 
If  he  spends  62 1%  of  his  salary,  and  saves  the  remainder, 
how  much  does  he  spend  ?     How  much  does  he  save  ? 

31.  An   auriferous   ore  contains    .5%    of   gold.     How 
many  pounds  avoirdupois  of  gold  would  2240  Ib.  of  this 
ore  yield  ? 

32.  A  copper  ore  contains  5J%  of  copper.      Find  the 
number  of  pounds  of  copper  in  500  Ib.  of  this  ore. 

33.  A  owns  16|%  of  a  boat  valued  at  $12,300.     What 
is  the  value  of  A's  share  of  the  boat  ? 


124       ADVANCED  BOOK  OF  ARITHMETIC 

34.  The  estimated  value  of  the  exports  of  the  United 
States  for  1899  was  $1,275,000,000.  The  percentages  of 
exports  from  United  States  ports  for  that  year  are  given 
as  follows:  New  York,  37.4%;  Boston,  10.43%;  Phila- 
delphia, 5.05%  ;  Baltimore,  8.9%;  New  Orleans,  6.4%; 
Galveston,  7.17%.  Find  the  value  of  the  exports  from 
these  cities. 

Given  a  number  as  a  per  cent  of  some  other  number, 
to  find  the  other  number. 

Example  1.  If  15.5%  of  a  number  equals  22.785,  what 
is  the  number  ? 

15.5%  of  the  number  =  22. 785 

22.785 


15.5 


1  %  of  the  number  = 


.-.  100  %  of  the  number  =  x  100  =  147. 

15.5 

/.  the  number  is  147. 

Example  2.    If  3|  %  of  a  number  is  2934,  what  is  the 
number  ? 


. 
100     400      80 

-g^j-  of  the  number  =  2934. 
.*.  -g1^  of  the  number  =  978. 
.-.  fj.  of  the  number  =  78,240. 
/.the  number  is  78,240. 

EXERCISE  75 

1.  If  5  %  of  a  number  equals  185,  what  is  the  number  ? 

2.  If  12|  %  of  a  man's  salary  is  $156,  what  is  the  man's 
salary  ? 

3.  If  37^  %  of  a  man's  property  is  valued  at  $324,  what 
is  the  value  of  his  property  ? 


PERCENTAGE  125 

4.  The  number  360  is  equal  to  5%  of  what  number? 
6%  of  what  number?  8%  of  what  number?  9%  of  what 
number  ? 

5.  The  number  120  is  equal  to  3%  of  what  number? 
4%  of  what  number?  5%  of  what  number?  6%  of  what 
number?  8%  of  what  number?  12%  of  what  number? 

6.  $750  is  6%  of  what  sum  of  money?  6|%  of  what 
sum  of  money?    7-|-%  of  what  sum  of  money?    12|%  of 
what  sum  of  money  ? 

7.  $108  is  30%  of  what  sum?    25%  of  what  sum? 
331%  of  what  sum?  44|%  of  what  sum? 

8.  $450  is  6^%  of  what  sum?   6|%   of  what  sum? 
8J%  of  what  sum?   12^%  of  what  sum?   16 f%  of  what 
sum  ? 

9.  $420  is  37^%  of  what  sum?  62|%  of  what  sum? 
87^%  of  what  sum? 

10.  In  1900  the  commercial  value  of  silver  was  3%  of 
the  value  of  gold.     Find  the  number  of  ounces  of  silver 
equivalent  in  value  to  126  ounces  of  gold. 

11.  Of  the  population  of  the  United  States  in  1900, 
27,849,760  were  married.     This  number  was  36.5%  of  the 
population.     Find  the  population  in  1900. 

12.  The  census  of  1900  gives  the  number  of  married 
men  in  the  United  States  as  14,003,798.     This  number 
was  35.9%  of  the  number  of  males.     Find  the  male  popu- 
lation. 

13.  The  census  of  the  same  year  gives  the  number  of 
married  women  in  the  United  States  as  13,845,963.     This 
number  was   37.2%  of  the   entire   number   of    females. 
Find  the  female  population. 


126  ADVANCED  BOOK  OF  ARITHMETIC 

14.  Of  the  number  of  illiterates  above  10  years  of  age  in 
the  United  States  15.5%,  according  to  the  census  of  1900, 
can  neither   read   nor  write.     This   number   is   955,840. 
Find  the  number  of  illiterates  above  10  years  of  age  in  the 
United  States  in  1900. 

15.  Of  the  average  value  of  the  raw  cotton  exported 
from  the  United  States  in  5  years  50.4%  went  to  Eng- 
land.    If  the  export  of  raw  cotton  to  England  amounts  in 
value  to  1107,500,000,  find  the  average  value  of  the  raw 
cotton  exported. 

16.  Of   the   sheep   exported  from  the   United    States 
86%  are  shipped  to  England.     If  the  value  of  the  sheep 
shipped   to   England  in  a  certain   year  was  $1,685,800, 
find   the   total   value   of   the   export   of    sheep   for   that 
year. 

17.  Of  the  pupils  attending  school  in  a  certain  city  .6% 
are  in  the  senior  class  of  the  high  school.     If  the  senior 
class  numbers  21,  find  the  number  of   pupils   attending 
school  in  that  city. 

18.  In  a  certain  city  the  number  of  pupils  promoted 
at    the    end    of    the    scholastic    year    was    2765.     This 
number  was   79%  of   the   number   of   pupils   in   school. 
Find    the    number   of   pupils   attending   school   in   that 
city. 

19.  The  foreign -born  population  of   a  city  is  10,668. 
This  number  is  3^%  of  the  population  of  the  city.     Find 
the  population  of  the  city. 

20.  Of  the  water  of  the  Dead  Sea  22.8?T%  is  saline 
material.     If  a  quantity  of  Dead  Sea  water  is  evaporated, 
and  the  saline  material  left  behind  weighs  914.28  lb.,  what 
is  the  weight  of  the  water  before  evaporation  ? 


PERCENTAGE  127 

To  find  -what  per  cent  one  number  is  of  another. 
Example  1.    What  per  cent  of  12  is  5  ? 
5  is       of  12. 


Example  2.  The  value  of  the  property  of  the  Fort 
Worth  and  Denver  City  Railroad  in  Texas,  as  ascertained 
by  the  Railroad  Commission  of  Texas,  for  the  year  1906 
was  $5,771,600,  and  the  income  from  its  operation  was 
$1,178,040.  Find  the  rate  per  cent  of  income  from  opera- 
tion. 

4=20.4,, 

To  find  what  per  cent  one  number  is  of  another,  find  what 
fraction  the  first  number  is  of  the  second  and  multiply  by 
100%. 

EXERCISE  76 

1.  What  per  cent  of  15  is  12? 

2.  What  per  cent  of  20  is  4  ?   is  7  ?   is  11  ?   is  13  ? 

3.  What  per  cent  of  40  is  2  ?   is  8  ?   is  12  ?   is  17  ? 

4.  What  per  cent  of  90  is  9  ?   is  12  ?   is  27  ?    is  11|  ? 

5.  What  per  cent  of  120  is  15  ?   is  18  ?    is  45  ?    is  80  ? 

6.  What  per  cent  of  480  is  28.8?    is  33.6?    is  40  ? 

7.  What  per  cent  of  1728  is  345.6  ?  is  155.52  ?  is  288  ? 

8.  What  per  cent  of  231  is  21  ?  is  77  ?  is  34.65  ? 

9.  What  per  cent  of  5280  is  88  ?  is  440  ?  is  330  ? 

10.  What  per  cent  of  a  bushel  is  a  quart  ? 

11.  What  per  cent  of  a  mile  is  8  rd.  ? 

12.  What  per  cent  of  an  acre  is  a  square  rod  ? 

13.  What  per  cent  of  a  chain  is  a  yard  ? 

14.  What  per  cent  of  1  sq.  ch.  is  1  sq.  rd,  ? 


128 


ADVANCED   BOOK  OF  ARITHMETIC 


15.  What  per  cent  of  1  gal.  is  1  pt.? 

16.  What  per  cent  of  1  sq.  rd.  is  7  sq.  yd.  5  sq.  ft. 
9  sq.  in.? 

17.  What  per  cent  of  a  rod  is  1  yd.  2  ft.  6  in.? 

18.  What  per  cent  of  1  mi.  is  1  knot? 

19.  What  per  cent  of  1  mi.  is  an  arc  of  1'  measured  on 
the  40th  parallel  of  latitude  ?     (!'  =  4670  ft.) 

20.  What  per  cent  of  1  mi.  is  22  yd.?  is  176  yd.? 

21.  What  per  cent  of  1  sq.  mi.  is  the  S.  W.  1  of  N.E. 
-|  of  a  section  of  land  ? 

22.  What  per  cent  of  a  common  year  is  73  da.  ?  is  219 
da.?  is  292  da.? 

23.  A  meter  is  39.37  in.     What  per  cent  of  a  meter  is 
1  yd.  ?     What  per  cent  of  1  yd.  is  1  meter? 

24.  A  kilometer   is    1000    meters.     What  per  cent  of 
1  mi.  is  1  km.? 

25.  What  per  cent  of  1  Ib.  avoirdupois  is  1  Ib.  troy? 

26.  What  per  cent  of  1  oz.  avoirdupois  is  1  oz.  troy? 

27.  What  per  cent  of  the  area  of  each  of  the  following 
states  consists  of  irrigated  land? 


ACRES  IRRIGATED 

AREA  IN  SQUARE  MILES 

(a)  California  . 

1,446,114 

158,360 

(b)  Colorado 

1,611,270 

103,925 

(c)  Louisiana  . 

201,685 

48,720 

(d)  Montana     . 

951,054 

146,080 

(e)  Nevada 

501,168 

110,700 

(/)  Oregon       . 

388,110 

96,030 

(</)  Utah  .... 

629,290 

84,970 

(Ji)  Washington 

135,470 

69,180 

(0   Wyoming  . 

605,230 

97,890 

PERCENTAGE 


129 


28.   Find  the  increase  per  cent  in  the  population  of  each 
of  the  following  cities  for  the  ten  years  from  1890  to  1900: 


POPULATION  IN  1890 

POPULATION  IN  1900 

Mobile   ..... 

31,076 

38,469 

Little  Rock    .... 

25,874 

38,307 

Los  Angeles  .... 

50,395 

102,479 

Denver  

106,713 

133,859 

Pensacola       .... 

11,750 

17,747 

Savannah       .... 

43,189 

54,244 

Springfield     .... 

24,963 

34,159 

Evansville      .... 

50,756 

59,007 

Dubuque        .... 

30,311 

36,297 

Kansas  City,  Kan. 

38,316 

51,418 

Lexington      .... 

21,567 

26,369 

Kansas  City,  Mo.  . 

132,716 

163,752 

Minneapolis 

164,738 

202,718 

29.  The  total  production  of  butter  in  the  United  States 
in  1899  is  estimated  at  1,430,000,000  Ib.    Of  this  quantity 
16,002,000  Ib.  was  exported.     Find  the  per  cent  of  the 
total  production  exported. 

30.  For  the  year  1899  the  total  production  of  cheese  in 
the  United  States  is  estimated  at  300,000,000  Ib.     Of  this 
quantity  27,203,200  Ib.  was  exported.     Find  the  per  cent 
of  cheese  exported. 

31.  The  number  of  farms  in  the  United  States  June  1, 
1900,  was  5,739,657.     The  number  of  farms  operated  by 
owners  was  3,713,371.     The  number  of  farms  operated  by 
tenants   was   752,920.      The   number  of  farms   operated 
by  share  tenants  was  1,273,366.     Find  the  per  cent  oper- 
ated by  owners,  by  tenants,  and  by  share  tenants  respec- 
tively. 


130  ADVANCED  BOOK  OF  ARITHMETIC 

COMMERCIAL  DISCOUNTS 

Commercial  or  trade  discount  is  an  allowance  made  upon 
the  list  price  of  goods,  or  on  the  amount  of  a  bill. 

Discounts  are  reckoned  upon  the  basis  of  100  ;  in  other 
words,  as  so  many  per  cent. 

Example.  If  watches  are  listed  at  $35  each,  and  a 
discount  of  10  %  is  allowed,  what  is  the  cost  of  one  of  the 
watches  ? 

SOLUTION.    10  %  of  $35  =  $3.50  =  the  discount. 
$35 -$3.50  =  $31. 50. 
.-.  the  cost  is  $31.50. 

EXERCISE   77 

Find  the  cost  when  the  list  prices  and  rates  of  discount 
are: 

1.  $60,  20%  off.  16.  $175,  2%  off. 

2.  $75,  30%  off.  17.  $213,  4%  off. 

3.  $15,  25%  off.  18.  $723,  6  %off. 

4.  $14,  20%  off.  19.  $3280,  8%  off. 

5.  $24,  10%  off.  20.  $2712,  7%  off. 

6.  $32,  15%  off.  21.  $5350,  8J%  off. 

7.  $68,  121%  Oif.  22.  $2490,  6-|%  off. 

8.  $78,  16|%off.  23.  $3778,  9%  off. 

9.  $92,  331%  off.  24.  $5062,  12%  off. 

10.  $80,  37|%  off.  25-  $885,  15%  off. 

11.  $152,  121%  off.  26.  $363, 12%  off. 

12.  $176,  25%  off.  27.  $689,  8%  off. 

13.  $88,  5%  off.  28.  $2034,  7%  off. 

14.  $96,  5%  off.  29.  $992,  7-|%  off. 
is.  $125,  15%  off.  30.  $2572,  4J%  off. 


COMMERCIAL  DISCOUNTS  131 

COMMERCIAL  DISCOUNTS  WHEN    TWO    OR    MORE   ARE 
ALLOWED 

In  some  cases  several  discounts  are  allowed.  If  there 
are  two  or  more  discounts,  the  first  is  reckoned  on  the  list 
or  catalogue  price  ;  the  next  is  reckoned  on  the  remainder 
after  deducting  the  first  discount ;  the  third  is  reckoned 
on  the  second  remainder  ;  and  so  on. 

Example  1.  What  is  the  cost,  if  the  list  price  is  $  750, 
and  discounts  of  20%,  15%,  and  10%  are  allowed? 

SOLUTION.    First  discount  =  20%  of  $750  =  |  of  $750 
=  $  150.     .-.  the  first  remainder  =  $ 750  -  $  150  =  $  600. 
The  second  discount  =  15  %  of  1600  =  $90. 

$600 -$90  =  $510. 

The  third  discount  =  10  %  of  $510  =  $51. 
$510 -$51  =  $459.       Or 
100  %  -  20  %  =  80  %  ;  100  %  -  15  %  =  85  % ; 

100%  -10%  =90%. 
90  %  of  85  %  of  80  %  of  $750  =  $459. 
Example  2.    What  single  discount  is  equivalent  to  the 
three  discounts  in  the  above  example  ? 

SOLUTION.  80%  x  85  %  x  90%  =  .8  x  .85  x  .9  =  .612 
=  61.2%. 

100  %  -  61.2  %  =  38.8  %  Ana. 

EXERCISE   78 

1.  A  suit  of  clothes  is  marked  $70,  and  is  sold  with 
discounts  of  25  %  and  10  %  for  cash.       Find  the  selling 
price. 

2.  On  a  bill  of  $  900  two  discounts  of  20  %  and  15  % 
are  allowed.     What  is  the  net  amount  of  the  bill? 

3.  If  goods   are   marked   $175  and  sold  for  $122.50, 
what  is  the  discount  ? 


132  ADVANCED  BOOK  OF  ARITHMETIC 

4.  On  a  bill  of  $1500  discounts  of  25 /o,  15  /o,  and 
are  allowed.     Find  the  net  cash  amount  of  the  bill. 

5.  A  piano  is  listed  at  $450,  with  discounts  of 
12J-^,  and  10/>.     Find  the  cost  price  to  the  purchaser. 

6.  Find  the  cash  value  of  a  bill  of  $320  with  discounts 
of  15/o,  10/o,  and  5/o. 

7.  Suppose  you  were  offered  a  single  discount  of  45/>, 
or  two  discounts  of  30  />  and  20  Jfc,  which  would  you  take  ? 
What  would  be  the  difference  in  a  bill  of  $1000  ? 

8.  A  dealer  buys  a  quantity  of  goods  marked  $550, 
with  a  discount  of  20/>.     If  he  sells  the  goods  at  6jfc  above 
the  marked  price,  what  is  his  gain  per  cent  ? 

9.  If  I  buy  goods  listed  at  $150,  writh  a  discount  of 
10/>,  and  sell  them  at  12  fi  above  the  marked  price,  find 
my  gain  per  cent. 

10.  A  bookseller  buys  100  books  marked  $1.50  each,  at 
a  discount  of  20 />  and  sells  them  at  the  marked  price. 
What  is  his  gain  per  cent  ? 

11.  A  bookseller  bought  100  books  at  $1.25  each.     He 
make  a  profit  of  20 />  after  giving  a  discount  of  ^.     At 
what  price  did  he  mark  each  book,  and  what  was  his  profit  ? 

12.  Find  the  cost  price  in  each  case,  if  the  list  price  and 
the  rates  of  discount  are  as  follows : 

LIST  PRICE  DISCOUNTS 

O)  $480  20/o,  80  #,  10 /o 

(6)  $1000  40/o,5^,4/o 

0)  $775  25/o,  I 

(i)  $880  12i/o,4/o 

00  $720  £,  16/o 

(/)  $960  12i?o,  12  Jb 

(#)  $1200  10^,  8/0,  5  Jfe 

Qi)  $1760 


PROFIT  AND  LOSS  133 

PROFIT  AND  LOSS 

In  actual  business,  gains  and  losses  are  reckoned  as  a 
per  cent  of  the  cost  price. 

Example  l.     How  much  does  a  person  gain  by  buying 
360   yd.    of   cloth   at   $1.30  per  yd.  and  selling  it  at  a 
profit  of  15  %  ?     What  is  the  selling  price  per  yard  ? 
SOLUTION.     360  x  $1.30  x  15  %  =  $70.20,  gain. 
$1.30 

.13    =10%  of  $1.30 
.065=    5%  of  $1.30 
$1.495  =  selling  price  per  yd. 
Observe,  selling  price  per  yd.  is  115  %  of  cost  price. 

Example  2.    A  dealer  buys  apples  at  $1.75  per  barrel, 
and  sells  them  at  $2.10  per  barrel.     Find  his  gain  per  cent. 
SOLUTION.     $2.10  -  $1.75  =  $.35. 

j|p~  =  1.     The  gain  is  £  of  the  cost.     |  =  20%. 

EXERCISE  79 

1.    Find  the  selling   price   of   articles,  the  cost   prices 
and  rates  per  cent  of  profit  being  given  as  follows  : 
COST  PRICE  RATE  PER  CENT  OP  PROFIT 

(a)  $150  12% 

(i)  $75  25% 

00  $31  7i% 

(cT)  $215  16|% 

O5  $540  27% 

GO  $318  S£% 

(<7)  $512  18|% 

(A)  $234  15% 

(0  $457 


134  ADVANCED  BOOK  OF  ARITHMETIC 

2.    Find  the  rate  per  cent  of  profit  or  loss,  if  the  cost 
prices  and  selling  prices  are  given  as  follows  : 

COST  PRICE  SELLING  PRICE 

(a)  $   .20  9  .25 

(5)  $   .22  $   .20 

0)  $  .90  $1.50 

(df)  $2.10  $1.40 

0)  $125  $160 

Given  selling  price  and  rate  per  cent  of  profit  or  loss,  to 
find  cost  price. 

Example  l.    A  piano  was  sold  for  $450  at  a  profit  of 
121  %  .     Find  the  cost  price. 

(100  %  +  12i  %)  of  cost  =  1121  cf0  of  cost  =  f  of  cost. 

|  of  cost  =  $450. 

Therefore,  cost  =$450  •*-  §  =  $400. 

Example  2.    A  dealer  sells  goods  for  $200.56  at  a  loss 
of  8%.     Find  the  cost  price. 

(100  %  -  8  %  )  of  cost  =  92  %  of  cost. 
92%  of  cost  =  $200.  56. 

...  cost  =  »~  1218.00. 


EXERCISE  80 

1.    Find  the  cost  price,  the  selling  price  and  rate  per 
cent  of  profit  being  given  as  follows  : 

SELLING  PRICE  RATE  PER  CENT  OF  PROFIT 

(a)  $63  25% 

(6)  1143  8£% 

O)  $54  8% 


(e)  $205 

(/)  $315  5% 


PROFIT  AND  LOSS  135 

2.  Find  the  cost,  if  the  selling  price  and  the  rate  per 
cent  of  loss  are  given  : 

SELLING  PRICE  RATE  PER  CENT  OF  Loss 

(a)                       $41.30  10% 

(8)'                    $87.50  20% 

(<?)                      $90  30% 

(d)                      $45.50  50% 

O)                      $59.90  331% 

(/)                      $33.60  16|% 

(#)                      $55.80  20% 

(Ji)                    $253  12|% 

Example  l.  A  man  sold  a  horse  for  $126,  thereby  los- 
ing 20%.  What  should  have  been  the  selling  price  to 
make  a  profit  of  15  %  ? 

SOLUTION.     The  selling  price  is  80  %  of  cost. 

To  make  a  profit  of  15%,  the  selling  price  should  be 
115  %  of  cost.  In  this  problem  80  %  of  a  number  is 
given  and  115  %  of  it  is  required.  Hence, 


115  x        r=  $181.125=  required  selling  price. 

80 

HJxample  2.  How  should  goods  be  marked  so  that  a 
dealer  may  give  a  discount  of  20  %  off  and  still  make  a 
profit  of  15  %  ? 

SOLUTION.  (100  %  —  20  %)  of  marked  price  =  80  %  of 
marked  price. 

Cost  +  15  %  of  cost  =  115  %  of  cost. 
%  of  marked  price  =  115  %  of  cost. 


1  %  of  marked  price  =         '°  of  cost. 
80 


.-.  100  %  of  marked  price  =          L  x  100  =  143|  %  of  cost. 

80 

The  goods  must  be  marked  43|  %  above  cost. 


136  ADVANCED  BOOK  OF  ARITHMETIC 

EXERCISE    81 

1.  A  sold  a  lot  for  $3835  at  a  gain  of  18%.     Find 
the  cost. 

2.  By  selling  a  piano  for  $270,  a  dealer  loses  10%. 
Find  the  cost  of  the  piano. 

3.  By  gaining  25  %  of  his  capital,  a  merchant  increases 
his  capital  to  $4550.     What  was  his  original  capital  ? 

4.  If  goods  are  bought  for  $20  and  sold  for  $22.50, 
what  is  the  gain  per  cent  ? 

5.  If  the  cost  is  $48  and  the  selling  price  is  $52,  what 
is  the  gain  per  cent  ? 

6.  If  the  cost  is  $190  and  the  selling  price  is  $152, 
what  is  the  loss  per  cent  ? 

7.  If  the  cost  is  $145  and  the  selling  price  is  $159.50, 
what  is  the  gain  per  cent? 

8.  If  the  cost  is  $118  and  the  selling  price  is  $128.62, 
what  is  the  gain  per  cent? 

9.  If  the  selling  price  is  $106.70  and  the  gain  is  10  %, 
what  is  the  cost  ? 

10.  If  the  selling  price  is  $75.25  and  the  loss  is  121  % 
what  is  the  cost  ? 

11.  If  the  selling  price  of  a  rug  is  $90  and  the  gain  is 
20  %,  what  is  the  cost  of  the  rug? 

12.  If  the  selling  price  is  $89.25  and  the  loss  is  15%, 
what  is  the  cost  ? 

13.  If  the  selling  price  is  $95  and  the  gain  is  18|%, 
what  is  the  cost?     What  is  the  profit  ? 

14.  By  selling  a  horse  for  $168,  a  man  gains  40%   of 
the  cost.     What  is  the  cost  of  the  horse  ? 

15.  By  selling  velvet  at  $4.55  a  yard,  a  clothier  makes 
a  profit  of  8-|  % .     Find  the  cost  per  yard  of  the  velvet. 


PROFIT  AND   LOSS  137 

16.  At  an  auction  a  dealer  buys  goods  at  20  %  below 
the  market  price.     If  he  sells  these  goods  at  the  market 
price,  what  is  his  gain  per  cent  ? 

17.  A  lawyer  collects  a  debt.     He  charges  3  %  for  col- 
lection, and  remits  to  his  client,  after  deducting  his  fee, 
$952.54.     Find  the  amount  of  the  debt. 

18.  If  property  was  sold  for  $11,778,  at  a  loss  of  2|%, 
what  was  the  value  of  the  property  ? 

19.  Two  horses  were  sold  for  $  200  each,  one  at  a  gain 
of  20  %  and  the  other  at  a  loss  of  20%  .     Did  the  seller  gain 
or  lose  by  the  transaction,  and  how  much  ? 

20.  A  sells  goods  to  B  at  a  profit  of  10%,  and  B  sells 
them  to  C  at  a  profit  of  15%.     If  C  paid  $253  for  the 
goods,  find  A's  cost  price. 

21.  A  merchant  increases  his  capital  18|%,  and  at  the 
end  of  a  second  year  he  also  increases  his  capital  18|  % . 
If  his  capital  is  then  $  14,440,  what  was  it  at  first  ? 

22.  The  first  year  A  adds  25  %  to  his  capital,  the  next 
year  he  adds  25  %  to  the  capital  of  the  previous  year,  the 
third  year  he  loses  40%  of  his  capital,  and  is  then  worth 
$10,800.     How  much  capital  did  he  begin  with  ? 

23.  If  it  takes  $81,415  for  the  running  expenses  of  the 
schools  of  a  city,  and  if  95  %  of  the  tax  levied  for  school 
purposes  goes  to  the  support  of  the  schools,  what  should 
be  the  amount  levied  for  school  purposes  ? 

24.  By  selling  tea  at  65^  per  pound  a  merchant  gained 
$118.75.     If  his  gain  was  62|-  %,  how  many  pounds  of  tea 
did  he  sell  ? 

25.  By  selling  turkeys  at  $2.50  each  a  dealer  makes  a 
profit  of  60%.     What  would  have  been  his  gain  per  cent 
had  he  sold  the  turkeys  for  $1.75  each? 


138  ADVANCED  BOOK  OF  ARITHMETIC 

26.  By  selling  wine  at  $2.10  a  gallon,  a  merchant  makes 
a  profit  of  20  %.     What  would  be  his  gain  per  cent  should 
he  sell  the  wine  for  $2  a  gallon  ? 

27.  A  merchant  mixes  tea  which  cost  him  60^  per  pound 
with  tea  which  cost  him  70^  per  pound  in  the  proportion 
of  5  Ib.  of  the  former  tea  to  6  Ib.  of  the  latter.     If  he 
sells  the  mixture  at  80^  per  pound,  find  his  gain  per  cent. 

28.  A  horse  is  sold  for  $145.50,  at  a  loss  of  3%.      At 
what  price  should  the  horse  have  been  sold  so  as  to  make 
a  profit  of  10  %  ? 

29.  If  oranges  are  bought  at  5  for  3^  and  sold  at  3  for 
5^,  what  is  the  gain  per  cent  ? 

30.  If  325  Ib.  of  sugar  are  bought  for  1 13,  at  what 
price  per  pound  must  it  be  sold  to  make  a  profit  of  25  %  ? 

31.  If  15  yd.  of  silk  are  bought  for  $26.25,  at  what 
price  per  pard  should  it  be  sold  to  make  a  profit  of  20  %  ? 

32.  By   selling   cloth   at   the   rate   of   17  J^  a   yard  a 
clothier  loses  12|%.     What  should  the  selling  price  per 
yard  be  so  as  to  make  a  profit  of  20  %  ? 

33.  A  merchant  buys  butter  at  18^  per  pound  and  sells 
120  Ib.  for  $25.20.     What  is  his  gain  per  cent  ? 

34.  A  grocer  buys  cheese  at  9^  per  pound  and  sells  it 
at  the  rate  of  8  Ib.  for  $1.     What  is  his  gain  per  cent  ? 

35.  A  merchant  mixes  two  kinds  of  tea  in  the  ratio  5  :  2. 
If  the  teas  are  worth  respectively  68  ^  and  75^  per  pound, 
what  should  be  the  selling  price  per  pound  so  as  to  make 
a  profit  of  20  %  ? 

36.  If  I  buy  a  horse  for  $80  and  sell  him  for  $71,  what 
is  the  loss  per  cent  ? 

37.  By  selling  a  horse  for  $83.30  I  lost  15%.     What 
did  the  horse  cost  ?  '. 


PERCENTAGE  139 

38.  Two  men,  A  and  B,  buy  two  horses,  each  paying  the 
same  price.     A  sells  his  horse  for  $90  at  a  loss  of  14|  %. 
B  sells  his  horse  at  a  loss  of  6  % .     Find  the  selling  price 
of  B's  horse. 

39.  A  merchant  buys  70  yd.  of  cloth  at  50  ^  a  yd.  and 
sells  it  at  a  gain  of  15  %.     He  buys  also  70  yd.  of  silk  at 
90  /  a  yard.     On  both  he  gains  10  %.     At  what  price  per 
yard  does  he  sell  the  silk  ? 

40.  I  buy  a  quantity  of  barley  and  intrust  it  to  an  agent 
to  sell.     The  agent  sells  it  at  an  advance  of  25  %  on  the 
cost,  and  after  deducting  a  commission  of  2%  he  remits 
the  balance,  $539.     How  much  did  the  barley  cost  me  ? 

41.  A  fruit  dealer  buys  a  crate  of  oranges  for  $2.50. 
He  sells  them  at  2/  each,  making  a  profit  of  60  %.     How 
many  oranges  are  in  the  crate  ? 

42.  If  goods  are  bought  at  25  %  below  the  retail  price 
and  sold  at  the  retail  price,  what  is  the  gain  per  cent? 

43.  A  coal  dealer  buys  120  T.  of  coal  at  $4  a  ton.     He 
sells  ^  of  the  coal  at  an  advance  of  25  %,  ^  of  it  at  an  ad- 
vance of  50%,  and  the  remainder  at  an  advance  of  10%. 
Find  his  entire  profit  and  his  gain  per  cent  on  the  coal. 

44.  (a)    How  should  goods  be  marked  so  as  to  make  a 
profit  of  12  %  after  deducting  20  %  from  the  marked  price  ? 

(&)  How  should  they  be  marked  so  as  to  make  a  profit 
of  17  %  after  deducting  10  %  from  the  marked  price  ? 

(e)  How  should  they  be  marked  so  as  to  make  a  profit 
of  20%  after  deducting  25%  from  the  marked  price? 

(d)  How  should  they  be  marked  so  as  to  make  a  profit 
of  10  %  after  deducting  10  %  from  the  marked  price  ? 

(e)  By  taking  30  %  off  the  marked  price  a  merchant 
neither  gains  nor  loses.     How  were  the  goods  marked  ? 


140  ADVANCED   BOOK  OF  ARITHMETIC 

COMMISSION  AND  BROKERAGE 

A  very  large  proportion  of  the  buying  and  selling  of 
the  produce  of  the  country  is  done  through  commission 
merchants  and  brokers.  A  commission  merchant  is 
usually  intrusted  with  the  goods  bought  or  sold.  A 
broker  merely  buys  and  sells. 

The  fee  which  a  commission  merchant  charges  for  his 
services,  or  which  an  agent  who  buys  and  sells  land,  or 
collects  rents,  charges,  is  called  a  commission.  The  fee 
which  a  broker  charges  is  called  brokerage. 

Commission  and  brokerage  are  usually  reckoned  as  a 
rate  per  cent  of  the  buying  price  when  goods  or  other 
commodities  are  bought,  or  of  the  selling  price  when  they 
are  sold. 

The  person  who  employs  another  person  to  buy,  to  sell, 
or  to  collect  money  is  called  a  principal.  The  person  who 
transacts  business  for  a  principal  is  called  an  agent.  Com- 
mission merchants  and  brokers  act  in  the  capacity  of 
agents. 

Example  l.  A  commission  merchant  sold  400  boxes  of 
oranges  at  $2.75  per  box,  charging  6%  commission. 
What  was  his  commission,  and  how  much  did  he  remit  ? 

The  price  of  400  boxes  of  oranges  @  $2.75  =  $1100. 
5  %  of  $1100  =  $55,  the  commission. 
$1100  -  $55  =  $  1045,  amount  remitted. 

Example  2.  A  commission  merchant  charges  $94.50 
for  selling  4500  bushels  of  wheat.  Rate  of  commission 
2|%.  What  is  the  price  of  wheat  per  bushel  ? 

21  %  =  ^.     Selling  price  =  $  94.50  x  40  -  $  3780. 
Selling  price  of  one  bushel  =  $||f§  =  $.84. 


COMMISSION  AND  BROKERAGE  141 

Example  3.  A  real  estate  agent  remits  to  his  principal 
$9788.75,  being  the  amount  of  the  sales  of  four  city  lots 
after  deducting  a  commission  of  4 J  % .  Find  the  selling 
price  of  the  four  lots  and  the  agent's  commission. 

SOLUTION.      The   selling   price  —  4 £  %    of   the   selling 
price  =  95-| -%  of  the  selling  price. 
95J  %  of  the  selling  price  =  $9788.75. 

ffi»  Q7QQ   7^ 

.-.  1  %  of  the  selling  price  =  — — -^  — 

t/O^ 

.-.  100  %  of  the  selling  price  =  $9788.75  x  100  _  $i0,250. 
.-.  $10,250  -  $9788.75  =  $461.25,  the  commission. 

EXERCISE  82 

1.  What  is  the  commission  for  selling  200  A.  of  land 
at  $40  an  acre,  the  rate  of  commission  being  4  %  ? 

2.  An  agent  charges  6  %  for  selling  real  eatate.     If  he 
sells  350  A.  of  land  at  $50  an  acre,  find  his  commission 
and  the  sum  he  remits  to  his  principal.     What  per  cent  of 
the  selling  price  does  the  principal  receive  ? 

3.  If  200  boxes  of  oranges  are  sold  at  $3.75  a  box  by  a 
commission  merchant  who  charges  6%,  what  is  the  com- 
mission ?     How  much  is  remitted  to  the  principal  ? 

4.  If  800  bu.  of  wheat  are  sold  on  a  commission  of  J^ 
per  bushel,  what  is  the  commission  ? 

5.  How   many  bushels   of   barley   at   45^   per   bushel 
must  a  commission  merchant  sell  at  2J  %  commission  to 
make  an  annual  salary  of  $1080  ? 

6.  If  480  bbl.  of  flour  at  $3.90  a  barrel  are   sold   on 
commission  at  a  rate  of  3%,  find  the  commission.     What 
is  the  net  amount  realized  from  the  sale  ? 


142  ADVANCED  BOOK  OF  ARITHMETIC 


7.  A  broker  sells  2500  Ib.  of  beef  at  12**  a  pound. 
What  is  his  brokerage  at  \\%  ? 

8.  Thirty-six  hundred   gallons  of  oil  are  sold  by  a 
broker  at  45^  per  gallon.     If  he  charges  3^%  brokerage, 
find  his  brokerage  and  the  amount  remitted. 

9.  A    commission   merchant   sells   1500  T.  of  hay  at 
$15  per  ton,  and  charges  5%  commission.     Find  his  com- 
mission.    How  much  does  he  remit  ?     What  per  cent  of 
the  selling  price  does  he  remit  ? 

10.  A  coal  dealer  receives  a  commission  of  90^  a  ton 
for  selling  anthracite  coal,  and  80^  a  ton  for  selling  bitu- 
minous coal.     If  he  sells  an  equal  quantity  of  each  kind 
of  coal,  -and  if  his  entire  commission  is  $1530,  how  many 
tons  of  each  kind  does  he  sell  ? 

11.  A  broker  charges  35^  for  selling  a  bale  of  cotton. 
How   many   bales    must   he   sell   to    realize   for   himself 
$295.40?     If  a  bale  contains  500  Ib.,  and  the  brokerage 
is  equivalent  to  |%,  find  the  price  of  cotton  per  pound. 

12.  A  commission  merchant  sells  1200  doz.  eggs  at  18  $ 
per  dozen.     Find  his  commission  at  8%. 

13.  If  a  commission  of  $70.98  is  paid  for  buying  40 
A.  of  land  at  $54.60  an  acre,  what  is  the  rate  per  cent 
of  commission  ? 

14.  If  $284  is  received  for  selling  grain  on  a  commission 
of  ^  per  bushel,  how  many  bushels  of  grain  were  sold  ? 

15.  If  an  agent  charges  3%  for  collecting  debts,  and  in 
one  month  his  commission  from  this  source  is  $126,  what 
is  the  amount  collected  ? 

16.  A  real  estate  agent's  commission  at  ^\%  is  $351. 
Find  the  amount  of  his  sales  and  the  amount  he  remits  to 
his  principal. 


COMMISSION  AND  BROKERAGE 


143 


17.  A  broker  buys  10,000  bu.  of  wheat  at  79|^,  and 
sells  the  wheat  the  next  day  at  T9-|^,  charging  ^  a  bushel 
for  buying  and  ^f  for  selling.     Find  his  commission. 

18.  An  agent  remits  to  his  principal  $9184.50  as  the 
net  proceeds  from  the  sale  of  12,000  bu.  of  wheat.     Find 
the  agent's  commission  if  he  charges  at  the  rate  of  2|-%. 
Find  also  the  selling  price  of  wheat  per  bushel. 

19.  After  deducting   a   commission    of   3%,  an   agent 
remits  $1813.90  from  the  sale  of  oats  at  46|^  per  bushel. 
How  many  bushels  of  oats  were  sold? 

20.  When  the  market   price  of  pork  is  13.6^  per  lb., 
how  many  pounds  can  be  bought  for  $520.20,  if  2%  is 
charged  for  brokerage  ? 

21.  How  many  acres  of  land  can  be  bought  for  $4635 
at  $  30  an  acre,  by  a  real  estate  agent  who  charges  a  com- 
mission of  3  %  ? 

22.  Find  the  commission  and  amount  remitted  to  the 
principal  on  the  sale  of  the  following  articles  : 


09 

1  urkeys 

1750 

lb. 

@ 

9  $% 

commission 

(J) 

Oranges 

275 

boxes 

@ 

$2. 

95 

5% 

commission 

00 

Apples 

580 

bbl. 

@ 

$1. 

85 

8% 

commission 

09 

Oysters 

390 

bbl. 

@ 

$1. 

45 

5% 

commission 

(0 

Potatoes 

1270 

bu. 

@ 

$1. 

05 

1\% 

commission 

C/) 

Celery 

560 

bunches 

@ 

$  . 

85 

10% 

commission 

(.9) 

Onions 

240 

bu. 

@ 

11. 

25 

8% 

commission 

(A) 

Beans 

960 

bu. 

@  $2.15 

5% 

commission 

(0 

Rice 

3500 

lb. 

@ 

*  5% 

commission 

0') 

Chickens 

984 

lb. 

@ 

4 

^  9% 

commission 

(*) 

Cotton 

287 

bales 

@ 

$49. 

3% 

commission 

(0 

Peaches 

315 

boxes 

@    $1.35 

9% 

commission 

144  ADVANCED   BOOK  OF  ARITHMETIC 


INTEREST 

By  amount  is  meant  the  sum  of  principal  and  interest. 

To  find  the  simple  interest  on  a  sum  of  money,  Multiply 
the  principal  by  the  rate  to  get  the  interest  for  1  year,  and 
this  product  by  the  time  expressed  in  years. 

Example  l.  Find  the  interest  and  the  amount  of  $1780 
for  5  mo.  at  7%. 

Solution:  11780  x  .07  x  ^  =  $51.92,  interest  for  5  mo. 
$1780  +  151.92  =  $1831.92,  amount. 

A  concrete  quantity  which  is  contained  an  exact  number 
of  times  in  another  concrete  quantity  is  called  an  aliquot 
part  of  that  quantity. 

Thus,  21  yd.  is  an  aliquot  part  of  10  yd.;  $2.75  is  an 
aliquot  part  of  $11. 

Example.  Find  the  interest  on  $780  for  1  yr.  2  mo. 
10  da.  at  7  %. 

SOLUTION  BY  ALIQUOT  PARTS. 

$780 

.07 

$54. 60  =  int.  for  1  yr. 
2  mo.  =  1  of  1  yr.        9.10  =  int.  for  2  mo. 
10  da.  =  I  of  2  mo.       1.52  =  int.  for  10  da. 

$65.22  =  int.  for  1  yr.  2  mo.  10  da. 

EXERCISE   83 
Find  the  interest  on  : 

1.  $700  for  1  yr.  at  3  %  ;  for  1  yr.  at  5  %  . 

2.  $278  for  1  yr.  at  6%;  for  1  yr.  at  7%. 

3.  $598  for  1  yr.  at  9%;   for  1  yr.  at  8  %. 

4.  $289  for  2  yr.  at  4  % ;   for  2  yr.  at  5  %.. 


INTEREST  145 

5.  $1000  for  1  yr.  3  mo.  at  6  %  ;  for  1  yr.  5  rno.  at  6  %. 

6.  $1200  for  1  yr.  4  mo.  at  8  %  ;  for  1  yr.  5  mo.  at  7  %. 

7.  $1500  for  lyr.  7  mo.  at  9%;  for  1  yr.  8  mo.  at  8%. 

8.  11600  for  2  yr.  3  mo.  at  5  %  ;  for  1  yr.  7  mo.  at  6  %. 

9.  $  2000  for  1  yr.  6  mo.  at  4  %  ;  for  1  yr.  9  mo.  at  5  % . 

10.  $3500  for  10  mo.  at  7  %\  for  7  mo.  at  8%. 

11.  $156.40  for  1  yr.  at  4J  %  ;  at  5|  %. 

12.  $185.50  for  1  yr.  at  4J  %  ;  at  7  %. 

13.  $375.60  for  5  mo.  at  6  %;  for  4  mo.  at  5%. 

14.  $928.40  for  2  mo.  at  8  %  ;  for  3  mo.  at  6%. 

15.  $735.60  for  3  mo.  at  1%  ;  for  2  mo.  at  8%. 

16.  $1200  for  2  mo.  at  8%  ;  for  5  mo.  at  6%. 

17.  $1350.50  for  5  mo.  at  5%  ;  for  4  mo.  at  7  %. 

18.  $393.80  for  7  mo.  at  4  %  ;  for  5  mo.  at  3  %. 

19.  $385.40  for  8  mo.  at  5  %  ;  for  7  mo.  at  6  %. 

20.  $934.54  for  9  mo.  at  8  %  ;  for  8  mo.  at  9  %. 

21.  $2713.64  for  10  mo.  at  9%  ;  for  7  mo.  at  8%. 

22.  $3800  for  11  mo.  at  7  %  ;  for  10  mo.  at  6%. 

23.  $2825  for  10  mo.  at  7%  ;  for  9  mo.  at  5%. 

24.  $2700  for  1  yr.  1  mo.  at  6  %  ;  for  14  mo.  at  5  %. 

25.  $3280  for  1  yr.  3  mo.  at  5  %  ;  for  8  mo.  at  4  %. 

26.  $4500  for  8  mo.  15  da.  at  6  %  ;  for  7  mo.  at  7  %. 

27.  $329.50  for  7  mo.  10  da.  at  8%. 

28.  $982  for  9  mo.  18  da.  at  6  %. 

29.  $545  for  10  mo.  25  da.  at  8  %. 

30.  $775.24  for  11  mo.  24  da.  at  1%. 


146  ADVANCED  BOOK  OF  ARITHMETIC 

Example.  Find  the  interest  on  $384.42  from  Jan.  11, 
1907,  to  April  30,  1907,  at  7  %. 

SOLUTION.  First,  find  the  difference  between  the  two 
dates. 

YR.     Mo.    DA. 

1907     4     30 

1907     1     11 

3     19 

$384.42 
.07 


$26.9094  =  int.  for    1  yr. 

3  mo.  =  J  of    1  yr.  "      6.727  =  int.  for    3  mo. 

15  da.  =  £  of    3  mo.        1.121  =  int.  for  15  mo. 

3  da.  =  1  of  15  da.  .224  =  int.  for    3  da. 

1  da.  =  1  of    3  da.  .075  =  int.  for    1  da. 

$8.147  =  int.  for    3  mo.  19  da. 
Ana.  $8.15. 

EXERCISE   84 
Find  the  interest  on : 

1.  $450  from  Jan.  12  to  April  18  at  6  %. 

2.  $783  from  Feb.  14  to  April  24  at  8%. 

3.  $2385  from  Mar.  6  to  Nov.  11  at  6%. 

4.  $3950  from  July  4  to  Dec.  6  at  6  %. 

5.  $4280  from  Aug.  31  to  Nov.  18  at  8%. 

6.  $7335  from  July  5  to  Sept.  14  at  7  %. 

7.  $3280  from  Jan.  8  to  July  5  at  4%. 

8.  $4592.40  from  Jan.  10  to  May  4  at  3%. 

9.  $384.75  from  Mar.  10  to  Aug.  14  at 

10.  $327.50  from  April  6  to  July  3  at  9  %. 

11.  $935  from  Jan.  1  to  April  30  at  4J%. 

12.  $3540  from  Jan.  10  to  Oct.  5  at  4|-%. 

13.  $1382.60  from  Feb.  8  to  Nov.  1  at  5 


INTEREST  147 

EXERCISE   85 

Find  the  amount  of : 

1.  $800  for  1  yr.  2  mo.  15  da.  at  4  %. 

HINT.    Find  the  interest  and  add  it  to  the  principal. 

2.  $580  for  3  yr.  3  mo.  at  6  %. 

3.  $750  for  10  mo.  15  da.  at  8%. 

4.  $327.60  for  11  mo.  12  da.  at  9%. 

5.  $326.54  for  8  mo.  8  da.  at  5%. 

6.  $739.90  for  5  mo.  11  da.  at  4%. 

7.  $843.90  for  6  mo.  14  da.  at  8%. 

8.  $325  for  9  me    12  da.  at  4-|-%. 

9.  $982  for  8  mo.  16  da.  at  5%. 

10.  $  375  for  7  mo.  14  da.  at  6  %. 

11.  $1280  from  Jan.  10  to  July  14  at  8  %. 

12.  $3580  from  Jan.  14  to  Aug.  18  at  8%. 

13.  $1500  from  March  11  to  July  19  at  6  %. 

14.  $4350  from  Aug.  14  to  Dec.  17  at  7  %. 

15.  $1200  from  Jan.  1  to  July  8  at  7%. 

16.  $480  for  9  mo.  15  da.  at  5%. 

17.  $800  for  6  mo.  18  da.  at  6%. 

18.  $550  for  3  mo.  15  da.  at  8%. 

19.  $650  for  5  mo.  12  da.  at  6%. 

20.  $850  for  8  mo.  15  da.  at  5  %. 

21.  $980  for  7  mo.  at  6%. 

22.  $2329  for  1  yr.  7  mo.  16  da.  at  5%. 

23.  $3278  for  2  yr.  10  mo.  11  da.  at  7  %. 

24.  $2594  for  1  yr.  5  mo.  10  da.  at  3%. 

25.  $978  for  1  yr.  1  mo.  20  da.  at  6%. 

26.  $1857  for  1  yr.  5  mo.  18  da.  at  5%. 

27.  $903.53  for  1  yr.  3  mo.  at  5  %. 


148  ADVANCED  BOOK  OF  ARITHMETIC 


MEASUREMENTS  —  SPECIFIC   GRAVITY 

The  ratio  of  the  weight  of  any  given  volume  of  a  sub- 
stance to  the  weight  of  an  equal  volume  of  another  sub- 
stance taken  as  a  standard  is  called  the  specific  gravity  of 
that  substance. 

The  standard  taken  for  solids  and  liquids  is  water. 
Specific  gravity,  then,  simply  means  how  many  times 
as  heavy  as  water  a  substance  is.  Thus,  cast  iron  is  7.21 
times  as  heavy  as  water,  and  hence  its  specific  gravity  is 
7.21.  Cork  is  about  one  fourth  as  heavy  as  water,  and 
hence  its  specific  gravity  is  |.  1  cu.  ft.  of  water  weighs 

1,000  oz. 

TABLE  OF  SPECIFIC  GRAVITIES 

Ash  ...     .84     Ebony     .     1.33     Steel  .  .  7.83  Clay     .     .     1.2 

Beech   .     .     .85     Glass.     .     2.89     Copper  .  8.95  Mercury  .  13.57 

Brass    .     .  8.40     Gold  .     .  19.26     Silver  .  10.47  Bar  iron  .    7.79 

Butter  .     .     .94     Granite  .    2.78    Lead  .  .  11.38  Platinum    21.5 
Ice     ...     .92 

Example.    Find  the  weight  of  28  cu.  in.  of  mercury. 
SOLUTION. 

-^-r  x  1000  oz.  ==  weight  of  cu.  in.  of  water. 

28 

x  1000  oz.  =  weight  of  28  cu.  in.  of  water. 

28 


.Ul 

12 

6 
6 

: 

A  1728  A 
13.57  x 

j    u/j.    —    WClgil 

7  x  1000  oz. 

94990  oz. 

94990 

432 
The  factors 

432 
of  432  are  12,  6,  and  6. 

7915.833 

1319.305 

L6  |219.88  oz. 

13  lb.,  11. 

88  oz. 

MEASUREMENTS  149 

EXERCISE   86 

1.  Find  the  weight  of  1  cu.  ft.  of  steel ;  1  cu.  ft.  of 
glass ;  1  cu.  ft.  of  clay. 

2.  Find  the  weight  of  1  cu.  in.  of  water.     Find  the 
weight  of  1  gal.  of  water. 

3.  Find  the  weight  of  1  bushel  measure  filled  with 
water. 

4.  A  cubic  foot  of  marble  weighs  2700  oz.     Find  the 
specific  gravity  of  marble. 

5.  A  cubic  foot  of  sea  water  weighs  64|  Ib.     Find  the 
specific  gravity  of  sea  water. 

6.  A  cubic  foot  of  goat's  milk  weighs  65  Ib.     Find 
how  many  times  as  heavy  as  water  goat's  milk  is. 

7.  The  mercury  in  the  barometer  exactly  counterbal- 
ances the  pressure  of  the  atmosphere.     If  the  barometer 
is  30  in.  high,  find  the  pressure  of  atmosphere  upon  every 
square  inch  of  surface. 

8.  A  swimming  pool  of  fresh  water  is  25  ft.  by  16  ft. 
and  5  ft.  deep.     Find  the  weight  of  the  water  it  contains. 

9.  A  block  of  granite  is  6  ft.  by  4  ft.  and  2  ft.  thick. 
Find  its  weight  in  tons. 

10.  Find  the  weight  of  1  cu.  in.  of  gold;  of  1  cu.  in. 
of  silver;   of  1  cu.  in.  of  platinum  ;  of  1  cu.  in.  of  lead. 

11.  A  block  of  ice  3  ft.  by  2  ft.  and  1  ft.  thick  weighs 
how  many  pounds  ? 

12.  What  is  the  weight  in  tons  of  1  cu.  yd.  of  clay  ? 

13.  Find  the  weight  of  the  butter  required  to  fill  a  box 
16  in.  by  9  in.  by  8  in. 

14.  A  cellar  is  18  ft.  by  12  ft.  and  8  ft.  deep.     Find 
the  weight  of  the  water  required  to  fill  it. 


150  ADVANCED   BOOK  OF  ARITHMETIC 

15.  Find  the  weight  of  the  air  in  a  hall  27  ft.  by  24  ft. 
and  15  ft.  6  in.  high.     1  cu.  ft.  of  air  weighs  .08073  Ib. 

16.  A  cubic   foot   of   coal  weighs   81|   Ib.     Find   the 
specific  gravity  of  coal. 

17.  How  many  cubic   inches  of   copper  weigh  just  as 
much  as  1  cu.  in.  of  platinum? 

18.  Find  the  weight  of  a  block  of  ebony  4  ft.  long,  9  in. 
wide,  and  8  in.  thick. 

19.  Find  the  weight  of  a  block  of  ash  12  in.  long,  8  in. 
wide,  and  6  in.  thick. 

20.  How  many  times  as  heavy  as  glass  is  mercury? 

21.  What  is  the  weight  of  a  bar  of  iron  2  in.  by  2  in. 
and  8  ft.  long  ? 

22.  Find  the  weight  of  a  beam  of  beech  wood  8  in.  by 
6  in.  and  12  ft.  long. 

Example  l.    How  many  sq.  yd.  are  in  1  sq.  mile  ? 

1  sq.  mi.  =  1760  x  1760  x  1  sq.  yd.  =  3,097,600  sq.  yd. 

Example  2.    A  lot  in  the  form  of  a  rectangle  contains 
16  A.,  and  is  48  rd.  wide.     Find  its  length. 

SOLUTION.    48  x  length  =  16  x  160.  (160  sq.  rd.=  1  A.) 


Therefore,  length  =  16  *  16Q  =        =  53J.     An*.    53£  rd. 

48  o 

Example  3.  A  rectangular  plot  of  ground  55  yd.  by  11 
yd.  produces  2  bu.  of  buckwheat.  How  much  will  1  A. 
produce  at  this  rate? 

SOLUTION.   55  x  11  sq,  yd.  produce  2  bu. 

2 

1  sq.  yd.  produces  -  —  bu. 
55  x  11 

4840  sq.  yd.  produce  4840  x   2   bu  =  16  bu. 

55  x  11 


MEASUREMENTS  151 

EXERCISE  87 

1.  The  area  of  a  rectangle  is  12,500  sq.  ft.  and  one 
side  is  100  ft.     Find  its  other  dimension. 

2.  The  area  of  a  rectangle  is  30^  sq.  yd.  and  its  length 
is  5J  yd.     Find  its  width. 

3.  A  lot  contains  16  A.,  and  its  length  is  64  rd.     Find 
its  width. 

4.  The  area  of  a  room  is  252  sq.  ft.,  and  its  length  is 
18  ft.     Find  its  width. 

5.  A  rectangular  piece  of  ground  15  yd.  by  12  yd. 
yields  6  bu.  of  potatoes.     Find  the  yield  of  an  acre. 

6.  A  rectangular  tract  of  land  30  yd.  by  11  yd.  yields 

2  bu.  of  oats.     At  this  rate  how  much  will  1  A.  yield  ? 

7.  A  rectangular  tract  of  land  50  yd.  by  15  yd.  yields 

3  bu.  of  corn.     At  this  rate  how  much  will  1  A.  yield  ? 

8.  The  base  of  a  triangle  is  84  yd.  and  its  area  is  840 
sq.  yd.     Find  its  height. 

9.  Find  the  altitude  of  a  triangle  having  for  base  34 
yd.  and  area  289  sq.  yd. 

10.  Given  the  area  of  a  triangle  1024  sq.  yd.  and  base 
64  yd.,  find  its  altitude. 

11.  The  area  of  the  walls  of  a  room  is  560  sq.  ft.  and 
the  dimensions  of  the  room  are  16  ft.  by  12  ft.     Find  the 
height  of  the  room. 

12.  The  width  of  a  rectangle  is  3.9  ft.  and  its  area  is 
17.55  sq.  ft.     Find  the  length. 

13.  The  area  of  a  hall  is  497  sq.  ft.,  its  length  is  71  ft. 
Find  its  width. 

14.  A  street  1  mile  long  contains  an  area  of  5J  acres. 
Find  the  width  of  the  street  in  feet, 


152  ADVANCED   BOOK  OF  ARITHMETIC 

EXERCISE   88 

1.  The  inside  dimensions  of  a  box  car  are  36  ft.  by  8  ft., 
and  8  ft.  6  in.  high.     Find  the  number  of  cu.  ft.  in  the 
car,  the  number  of  bushels  it  will  hold  allowing  1^  cu.  ft. 
to  the  bushel,  and  the  weight  of  the  carload  of  corn. 

SOLUTION.    36  x  8  x  8|-  =  2448  =  number  cu.  ft.  in  car. 
2448  -r-  1|  =  2448  x  .8  =  1958.4  =  number  bu.  in  car. 
1958.4  x  56  Ib.  =  109670.4  Ib.  =  weight  of  corn  in  car. 

2.  The  dimensions  of  a  box  car  are  32  ft.  by  8  ft.  and  7 
ft.  high.     Find  the  number  of  cu.  ft.  in  the  car,  the  num- 
ber of  bushels  it  will  hold  allowing  1^  cu.  ft.  to  the  bushel, 
and  the  weight  of  the  carload  of  oats. 

3.  A  refrigerator  car  is  28  ft.  9  in.  long,  7  ft.  6  in.  wide, 
and  8  ft.  high.     Find  the  number  of  cu.  ft.  in  it.     The 
capacity  of   this   car   is    64,000  Ib.     How   many  dressed 
turkeys,  averaging  12|  Ib.  each,  will  the  car  hold,  allow- 
ing 4000  Ib.  for  ice  ? 

4.  A  car  100,000  Ib.  capacity,  length  40  ft.,  width  8  ft. 
6  in.,  height  8  ft.,  inside  dimensions,  is  loaded  with  wheat. 
Find  the  number  of  cu.  ft.  in  the  car.     Find  the  weight 
of  the  wheat  that  will  fill  it.    What  fraction  of  this  weight 
is  the  capacity  of  the  car  ? 

5.  A  car  40  ft.  long,  8   ft.  6  in.  wide,  8  ft.  high,  is 
loaded  with  4-ft.  wood.     How  many  cords  does  this  car 
contain?     What  is  the  value  of  the  wood  at  $4.75  per 
cord?     What  will  it  cost  to  ship  the  wood  from  Sublime 
to  San  Antonio  at  $1.50  per  cord? 

6.  The  electric  railroad  from  Rochester  to  Avon,  New 
York,  is  19  miles  long.     The  rails  used  in  its  construc- 
tion weigh  80  Ib.  to  the  yard.     Find  in  tons  the  weight  of 
the  rails  of  this  railroad. 


MEASUREMENTS  153 

EXERCISE   89 

Example  1.    Find  the  weight  in  pounds  avoirdupois  of 
5000  silver  dollars.     A  silver  dollar  weighs  412|  grains. 
SOLUTION.    1  dollar  weighs  41 2|  gr. 

5000  dollars  weigh  5000  x  412|  gr. 
=  ^00^±12i- lb_ 
7000 

2.  Find  the  weight  in  pounds  troy  of  1,000,000  silver 
dollars.     Find  also  the  weight  in  pounds  avoirdupois. 

3.  A  silver  dollar  is  90  %  silver.     Find  the  weight  of 
pure  silver  in  one  silver  dollar. 

4.  The  commercial  value  of  a  silver  dollar  is  the  value 
of  the  pure  silver  it  contains.     What  is  the  commercial 
value  of  a  silver  dollar  when  pure  silver  is  worth  66^  per 
oz.   (troy  oz.)? 

5.  A  10-dollar  gold  piece  weighs  258  grains.     Find  the 
weight  of  10,000  dollars  in  gold.     Find  also  the  weight  of 
1,000,000  dollars  in  gold. 

6.  Gold  coins  contain  90  %  pure  gold.    Find  the  weight 
of  pure  gold  in  a  10-dollar  gold  piece. 

7.  The  alloy  in  a  gold  coin  neither  adds  to  nor  takes 
from  the  value  of  the  coin.     Compute  the  value  of  480 
grains  of  pure  gold.     (Remember  that  90  %  of  258  gr.  of 
gold  is  worth  $10.) 

8.  The  bullion  value  of  the  pure  silver  in  a  silver  dollar 
at  the  average  price  of  silver  for  the  year  1905  was  $.472. 
Find  the  value  of  1  oz.  troy  of  pure  silver. 

9.  The  value  of  a  fine  ounce  of  silver  for  the  year  1906 
was  $.6769,  and  the  value  of  a  fine  ounce  of  gold  the  same 
year  $20.67.     Find  the  commercial  ratio  of  the  value  of 
gold  to  the  value  of  silver  for  that  year. 


154  ADVANCED  BOOK  OF  ARITHMETIC 

RATIO 

The  first  number  in  a  ratio  is  called  the  antecedent,  and 
the  second  number  is  called  the  consequent. 

Since  a  ratio  is  a  quotient,  we  may  multiply  antecedent 
and  consequent  by  the  same  number  without  affecting  the 
value  of  the  ratio.     It  follows  also  that  the  terms  of  a  ratio 
must  be  both  abstract,  or  both  concrete,  and  that  the  value 
of  a  ratio  is  always  an  abstract  number. 
Example  l.    Divide  343  in  the  ratio  4  :  3. 
SOLUTION.     Divide  343  into  7  equal  parts  and  put  4  of 
these  parts  in  one  group  and  3  of  them  in  another. 
.-.  first  part      =  f  of  343. 
.*.  second  part  =  f  of  343. 

Example  2.    Divide  136  into  parts  proportional  to  2,  3, 4. 
SOLUTION.     Divide  into  9  equal  parts  and  group,  as  in 
Example  1.  2  +  3  +  4  =  9. 

.  • .  first  part      =  f  of  136  =  30f . 
.  • .  second  part  =  f  of  136  =  45£. 
.  • .  third  part    =  |  of  136  =  60$. 
Example  3.    Divide  529  proportional  to  the  numbers  f, 

i.A- 

SOLUTION.  Multiply  each  of  the  fractions  by  16,  the 
L.  C.  M.  of  8,  2,  and  16.  The  results  are  6,  8,  and  9.  The 
number  may,  therefore,  be  divided  proportional  to  the 
numbers  6,  8,  and  9. 

6  +  8  +  9=23. 

.  • .  first  part      =  /%  of  529  =  138. 
.  •.  second  part  =  fa  of  529  =  184. 
. •.  third  part    =  fa  of  529  =  207. 
CHECK.        138  + 184  +  207  =  529. 

138  +  f  =  184  -  i  =  207  -f-  T9B. 


RATIO  155 

EXERCISE   90 

1.  Divide  $130  between  two  persons  in  the  ratio  8:5. 

2.  Divide  $985  in  the  ratio  8  : 2. 

3.  Divide  $3240  in  the  ratio  4  : 5. 

4.  Divide  $7128  proportional  to  1,  2,  3. 

5.  Divide  $7225  proportional  to  4,  6,  7. 

6.  Divide  2916  proportional  to  7,  9,  11. 

7.  Divide  4745  proportional  to  5,  6,  7,  8. 

8.  Divide  6728  proportional  to  13,  14,  15,  16. 

9.  Divide  5040  proportional  to  21,  22,  23,  24. 

10.  Divide  2592  proportional  to  15,  17,  19,  21. 

11.  Divide  95  in  the  ratio  | :  1. 

12.  Divide  1736  in  the  ratio  J  :-J. 

13.  Divide  365  in  the  ratio  21 :  3f . 

14.  Divide  1521  in  the  ratio  7|  :  4f 

15.  Divide  4225  proportional  to  1^,  If,  2J. 

16.  Divide  2189  proportional  to  1£,  2£,  3yL. 

17.  A,  B,  and  C  enter  into  partnership,  A  contributing 
$2450,    B,    $3500,    and   C,    $4000.      They  gain  $1990. 
Find  each  person's  share  of  the  profits,  supposing  each 
person's  gain  is  proportional  to  his  investment. 

18.  Divide  $2700  among  A,  B,  and  C,  so  that  A  shall 
receive  $4  as  often  as  B  receives  $5  and  C  receives  $6. 

19.  Three  men  rent  a  ranch  for  $900.     The  first  man 
puts  in  100  head  of  cattle,  the  second  85,  and  the  third 
115.     How  much  rent  should  each  pay  ? 

20.  Two   men    buy  two  lots  for  $10,000.     One   man 
pays  25%   more  for  his  lot   than  the  other  man   pays. 
How  much  does  each  pay? 


156  ADVANCED  BOOK  OF  ARITHMETIC 

PROPORTION 

Two  equal  ratios  constitute  a  proportion. 

Thus,  |  =  ^  or  2  :  3  =  6:  9  is  a  proportion. 

The  first  and  fourth  terms  of  a  proportion  are  called 
extremes;  the  second  and  third  are  called  means. 

The  law  of  a  proportion  is:  The  product  of  the  extremes 
equals  the  product  of  the  means. 

The  simplest  method  of  working  examples  in  proportion 
is  the  analytical  or  the  unit  method. 

Example  1.  If  17  bu.  of  wheat  cost  $15.30,  find  the 
cost  of  15  bushels. 

SOLUTION.     17  bu.  cost  $15.30. 

$15.30 
1  bu.  costs  — ij-y — 

15  bu.  cost  15  x®15,80  =$13.50. 

Example  2.    If  25  men  do  a  piece  of  work  in  18  days, 
how  long  will  it  take  20  men  to  do  the  same  work  ? 
SOLUTION.     25  men  take  18  da.  to  do  the  work. 

.•.1  man  takes  18  da.  x  25  to  do  the  work. 

.  •.  20  men  take  ^  of  18  da.  x  25  =  22 J-  days. 

EXERCISE  91 
Solve  analytically: 

1.  If    19    bu.  of  wheat  cost  $14.25,  find  the  price  of 
11  bu.  of  wheat. 

2.  Seven  men  can  dig  a  treaich  in  16  da.      How  long 
will  it  take  10  men  to  do  the  same  work  ? 

3.  If  11  sheep  cost  $71.50,  find  the  cost  of  17  sheep  at 
the  same  rate. 

4.  If  14  men   can  do  a  piece  of  work  in   11  da.,  how 
long  will  it  take  21  men  to  do  the  same  work  ? 


PROPORTION  157 

5.  The   earth   revolves   on   its   axis   15°    in   1    hour. 
Through  how  many  degrees  does  it  revolve  in  23  minutes? 

6.  Twelve  horses  plow  a  field  of   47  acres  in  7  days. 
How  many  acres  will  8  horses  plow  in  the  same  time? 

7.  A  pole  30  ft.  high  casts  a  shadow  24  ft.  long.    Find 
the  length  of  the  shadow  cast  by  a  pole  70  ft.  high. 

8.  How  long  will  it  take  20  men  to  pave  a  street 
which  15  men  pave  in  15  days? 

9.  A  sum  of  money  yields  $40.30  interest  in  125  days. 
What  interest  will  the  same  sum  yield  in  75  days? 

10.  A  train  runs  40  miles  in  1  hr.  and  20  min.     How 
far  will  it  run  in  2  hr.? 

11.  If  3|  acres  of  land  are  worth  $259,  what  is  the 
value  of  2J  acres? 

12.  If  |  of  an  acre  of  land  is  worth  $89,  how  much  is  a 
tract  measuring  2^  acres  worth? 

13.  A  train  runs  45  miles  an  hour.     Find  how  many 
feet  it  will  go  in  1  minute ;  in  1  second. 

14.  When  the  rate  of  a  train  is  36  miles  an  hour,  what 
is  its  rate  in  feet  per  second? 

15.  Sound  travels  1  mile  in  5  seconds.     How  long  will 
it  take  sound  to  travel  4400  feet? 

16.  Light  travels  from  the  sun  to  the  earth,  a  distance 
of  92,790,000  miles,  in  8  minutes  and  18  seconds.     Find 
its  rate  per  second. 

17.  If  |  of  a  clerk's  yearly  salary  is  $900,  what  is  his 
salary  per  month? 

18.  If  11 J  yards  of  carpet  can  be  bought  for  a  certain 
sum  of  money,  how  many  yards  can  be  bought  for  the 
same  sum  when  the  price  of  carpet  falls  8  %  ? 


158  ADVANCED  BOOK  OF  ARITHMETIC 

REVIEW 
EXERCISE   92 

1.  Multiply  the  sum  of  J,  ^,  |,  by  1T2^,  and  divide  the 

21 

product  by  ^* 
% 

2.  What   number   taken    from    20^   leaves   as    a   re- 
mainder 4|  ? 

3.  The  difference  between  two  numbers  is  1 J,  and  the 
less  is  4|.     Find  the  other  number. 

4.  Which  is  the  greater,  f  of  4$,  or  f  of  4-f  ? 

5.  From  £  of  the  sum  of  J,  |,  |-,  ^-,  take  the  sum  of 

A>  iV  6T- 

6.  From  £  of  (1  _  i  + 1)  take  ( J  +  |  -  £). 

7.  Express  in  pounds  the  difference  between  .0125  of  a 
ton  and  |  cwt. 

8.  What  fraction  having  48  for  denominator  is  equiva- 
lent to  .1875? 

9.  Find  in  feet  the  value  of  l|  of  a  mile. 

10.  Express  f  yd.  as  the  decimal  of  a  rod. 

11.  Express  f  of  95  Ib.  as  the  decimal  of  1^  cwt. 

12.  Express  27|  Ib.  as  the  decimal  of  a  ton. 

13.  Take  -|  T.  from  1J  T.,  and  express  your  result  in 
pounds. 

14.  From  |  of  a  right  angle  take  33°  45'. 

15.  From  ^  of  a  circumference  take  ^  of  the  circum- 
ference and  express  your  answer  in  degrees. 

16.  There   are   two   numbers  in  the  ratio  of  3  to  5. 
What  fraction  of  their. sum  is  their  difference? 


REVIEW  159 

17.  An  estate  is  left  to  A,  B,  and  C.     A  gets  |-  of  the 
estate,  B,  ^  of  the  estate,  and  C,  the  remainder.     What 
part  of  the  estate  does  C  get  ?     If  C's  share  is  $450,  what 
is  the  value  of  the  estate  ?     Find  A's  share  and  B's  share. 

18.  A  man  spends  f  of  his  salary  on  board,  ^  on  cloth- 
ing, -^  on  rent.     He  saves  the  remainder,  amounting  to 
$125.     Find  his  salary. 

19.  Subtract  -|  from  2.1,  and  divide  the  remainder  by 
.25. 

20.  The  dividend  is  3.562  and  the  quotient  is  .3125. 
Find  the  divisor. 

21.  Express  $5.24  as  a  decimal  of  $100. 

22.  After  giving  away  |,  -j^,  and  J£  of  his  money,  a 
man  has  left  $392.95.     How  much  money  had  he  at  first  ? 

23.  The  third  part  of  a  number  exceeds  the  fifth  part 
of  the  same  number  by  15.     What  is  the  number  ? 

24.  The  sixth  part  and  the  eighth  part  of  a  number 
together  make  66 J.     What  is  the  number? 

25.  If   |  and  ^  of  a  farm  are  together  worth  $1650, 
what  is  |  of  the  remainder  worth? 

26.  Subtract  the  product  of  |  and  f  from  their  sum. 

27.  What   number   divided  by   3|  gives  ly1^  for  the 
quotient  ? 

28.  Find  the  value  of  %  of  1^  of  $19.80. 

29.  A  man  owns  J-  of  a  boat  and  sells  f  of  his  share 
for  $750.     At  this  rate,  find  the  value  of  the  boat. 

30.  If  6  men  do  a  piece  of  work  in  9  days,  how  long 
will  it  take  4  men  to  do  the  same  work  ? 

31.  If   15   men   pave    a   street  in   16  days,  how  long 
will  it  take  40  men  to  pave  the  same  street  ? 


160  ADVANCED  BOOK  OF  ARITHMETIC 

32.  If  18  men  remove  an  embankment  in  12  da.,  how 
long  will  it  take  24  men  to  remove  the  embankment  ? 

33.  Two  trains  start  at  the  same  time  from  two  sta- 
tions 840  mi.  apart,  and  travel  toward   each   other,  one 
train  going  at  the  rate  of  35  mi.  an  hour,  and  the  other 
of  25  mi.  an  hour.      In  how  many  hours  will  they  meet  ? 

34.  If  a  man  performs  ^  of  a  piece  of  work  in  15  da., 
in  how  many  days  more  will  he  complete  the  work  ? 

35.  Add  |,  -£,  •§,  -j6g,  and  ^.      Express  the  sum  as  a 
decimal.     Check  by  reducing  to  decimals  and  adding. 

36.  What    is    the    smallest    number  which,    added   to 
the  sum  of   ^,  ^,  and  -|,  will  make  the   final  result  an 
integer  ? 

37.  If  -|  of  a  barrel   of  sugar  is   sold   and   afterward 
40  Ib.  are  sold,  how  many  pounds  of  sugar  were  in  the 
barrel  originally,  supposing  it  still  contains  90  Ib.? 

38.  By  selling  a  piano  at  -^  of  its  cost,  a  dealer  loses 
$98.     Find  the  cost  of  the  piano. 

39.  If  45  sq.  rd.  of  land  cost  $  18,  find   the   cost    of 
1  A.     Find  also  the  cost  of  3|  A. 

40.  If  |  of  a  clerk's  salary  per  year  is  $675,  find  his 
salary  per  month.     If  his  expenses  average   $48.89   per 
month,  how  much  will  he  save  per  year  ?     How  long  will 
it  take  him  to  save  $652.75  ? 

41.  If  31  A.  of  land  are  worth  $119,  find  the  value 
of  2|-  A.     How  much  is  a  rectangular  strip  of  this  land 
|  of  a  mile  long  and  33  feet  wide  worth  ? 

42.  How  much  is  a  plot  of   ground  80  ft.  by  60   ft. 
worth,  if  an  acre  is  worth  $55  ?     If  an  acre  is  worth  $121  ? 


REVIEW  161 

43.  Find  the  weight  of  a  piece  of  coal  in  the  shape  of  a 
rectangular  solid,  if  its  dimensions  are  1J  ft.  by  1J  ft.  by 
10  in.     A  cubic  foot  of  coal  weighs  81-|  Ib. 

44.  A  man's  property  is  assessed  at  $4550.     If  he  pays 
40^   on   every  $100   for  school   tax,  how   many   dollars 
school  tax  does  he  pay? 

45.  How  much  taxes  will  be  paid  on  real  estate  worth 
$9580,  if  the  tax  is  at  the  rate  of  $1.27  on  $100? 

46.  A  man  invests  $4800  and  gains  $540.     How  much 
does  he  gain  on  every  dollar  invested?     How  much  does 
he  gain  on  every  $100? 

47.  If   an   investment    of   $9600   produces   a    gain    of 
$1056,  find  the  gain  on  $1 ;  also  on  $100. 

48.  When  $8400  produces  a  profit  of  $1092,  how  much 
does  $  1  produce  ?     $  100  ? 

49.  The  school  tax  in  a  city  is  2  mills  on  the  dollar. 
The   assessed   valuation  of  the  property  is  $17,294,000. 
Find  the  total  tax  levied  for  school  purposes.     If  .95  of 
this  total  is  collectible,  find  the  amount  3ollected. 

50.  Find  the  tax  on  $33,254,000  at  4  mills  on  the  dollar. 

51.  A  man  insures  his  dwelling  for  $5450.     If  he  pays 
$11.50  on  every  $1000,  how  much  does  he  pay  altogether? 

52.  Find  the  insurance  on  a  house  valued  at  $7840,  if 
the  rate  of  insurance  is  $1J  on  every  $100? 

53.  If  $47.50  is  paid  to  insure  a  boat  valued  at  $9500, 
how  much  is  paid  on  $1?  on  $100. 

54.  The  distance  from  New  York  City  to  Plymouth, 
England,   is    2962   knots.       The    steamship    DeutscUand 
sailed  from  Plymouth  to  New  York  in  July,  1900,  in  5 
da.   15  hr.   45  min.     Find,  in  knots,  its  rate  per  hour. 
Find  also  its  rate  in  miles  per  hour. 


162  ADVANCED  BOOK  OF  ARITHMETIC 

55.  In  March,    1902,  the   steamship   La  Savoie   made 
the  voyage  from  Havre  to  New  York  in  6  da.  10  hr.     The 
distance  from  Havre  to  New  York  is  3170  knots  nearly. 
Find,  in  knots,  the  rate  per  hour.     Express  the  rate  also 
in  miles  per  hour. 

56.  The  steamer  Kronprinz    Wilhelm  made,  in  Septem- 
ber, 1902,  a  voyage  from  Cherbourg  to  New  York  in  5  da. 
11  hr.  57  min.     Find  its  rate  per  hour,  the  distance  from 
Cherbourg  to  New  York  being  3184  knots. 

57.  In  May,  1900,  a  passenger  train  ran  from  Burling- 
ton to  Chicago,  205.8  mi.,  in  3  hr.  8  min.  30  sec.     Find 
its  rate  per  hour. 

58.  The  fastest  time  on  record  by  a  passenger  train  for 
a  distance  over  450  miles  was  made  in  October,  1895,  on 
the  Lake  Shore  and  Michigan  Southern  Railroad,  from 
Chicago  to  Buffalo,  a  distance  of  510  mi.,  in  8  hr.  1  min. 
Find  its  rate  per  hour. 

59.  The  run  from  London  to  Edinburgh,  393  J  mi.,  has 
been  made  in  7  hr.  45  min.     Find  the  speed  per  hour. 

60.  The    market    quotations,    Feb.    19,    1903,    were  : 
wheat,    78|^   per  bushel;  corn,  45 1^   per  bushel;  oats, 
34^  per  bushel.     Find  the  price  of  100  Ib.  of  each  of 
these  commodities. 

61.  Market  quotations  of  live  stock  sales  are  in  dollars 
per  100  Ib.     Find  the  cost  of  : 

(a)  44  cattle,  average  weight  1121  Ib.,  @  $4.80. 

(6)  132  cattle,  average  weight  1018  Ib.,  @  $4.80. 

O)  24  cattle,  average  weight  915  Ib.,  @  $4.20. 

(d)  23  cattle,  average  weight  1060  Ib.,  @  $3.90. 

(e)  133  heifers,  average  weight  862  Ib.,  @  $4.00. 
(/)  69  calves,  average  weight  201  Ib.,  @  $5.00. 
(jgf)  82  hogs,  average  weight  188  Ib.,  @  $6.20. 


CHAPTER  III 

GENERAL  REVIEW  BY  TOPICS      r 
ADDITION 

To  add  numbers  is  to  find  a  single  number  equivalent 
to  the  numbers  jointly.  The  result  is  the  sum. 

Only  numbers  of  the  same  kind  can  be  added.  Thus, 
5  yd.  and  7  yd.  may  be  added;  but  5  yd.  and  $ 7  cannot 
be  added. 

5  ft.  and  7  in.  may  be  added   provided   the   5  ft.  is 
changed  to  inches,  or  the  7  in.  changed  to  feet.     The  sum 
in  one  case  is  67  inches,  in  the  other  case,  5^  feet. 
Example  1.    Add  279,  514,  928,  763. 

The  process  is  3,  11,  15,  24 ;   24  units  =  2  tens 
279  "  and  4  units. 

514         Write  the  4  units  and  carry  the  2  tens. 
928         2,  8,  10,    11,    18;    18   tens=l  hundred  and   8 
763     tens. 
2484         Write  8  and  carry  1. 

1,  8,  17,  22,  24 ;  write  24. 
Example  2.    Add  $2.79,  15.14,  19.28,  $7.63. 
12.79 
5.14 

9.28         The  process  is  the  same  as  in  Example  1. 
7.63 
$24.84 

163 


164 


ADVANCED  BOOK  OF  ARITHMETIC 


Example  3.    Add  6  ft.  8  in.,  3  ft.  6  in.,  5  ft.  4  in. 

FT.          IN. 

6        8  The   process   is  4,  10,  18 ;    18  in.  =  1  ft. 

366  in.     Write  6  in.,  carry  1  ft. 
5       4        1,  6,  9,  15.     Write  15  ft. 


15        6 


^ 


Example  4.    Add  6f  ,  3J,  51. 

Change  f  ,  |,  and  J  to  equivalent  fractions  having 
12  for  denominator. 

21      l_8  +  6+4_18 

-  3+2+3~     ~12~     "12        *' 

152         Write  |-,  carry  1.     1,  6,  9,  15.     Write  15. 

Observe  the  same  principle  pervades  the  four  examples. 
Units  of  the  same  name  are  placed  in  the  same  column, 
and  the  columns  added  separately. 

An  example  in  addition  may  be  checked  by  adding  the 
columns  in  reverse  order. 


EXERCISE  93 


Add: 

(i) 

(2) 

(3) 

00 

(5) 

4792799 

7998992 

5998476 

5879824 

6876894 

8399384 

8409499 

9208503 

9473473 

9219479 

7207383 

7294792 

8392393 

7777888 

7328337 

9476583 

8514798 

6555777 

6456789 

8474494 

4728737 

9998777 

7918924 

9875874 

9318327 

9219777 

6666784 

5729998 

8294295 

6438444 

6444673 

8542728 

6329888 

7318392 

7473478 

8299299 

5293294 

7774673 

9299497 

9888777 

9218288 

4445454 

6428427 

6713729 

6666679 

7666729 

7778898 

9444779 

9873876 

9218289 

ADDITION 


165 


6.    Add   vertically   and   horizontally,  arid   finally  sum 
the  vertical  and  the  horizontal  totals  : 


392 
876 
543 

878 
929 


965 
329 
707 
538 

928 


873 


393 
427 
925 


599 
307 
448 
388 
394 


222 
399 
937 
542 
234 


7.  The  mileage  of  railroads  in  operation  in  the  several 
states  is  given  for  the  years  1903,  1904,  and  1905  as 
follows : 


GROUP  AND  STATE 

1903 

1904 

1905 

NEW  ENGLAND 

Maine      ...... 

2,004.79 

2,029.89 

2,091.12 

New  Hampshire      .... 
Vermont          ..... 
Massachusetts          .... 
Rhode  Island           .... 
Connecticut    .         .         . 

1,191.42 
1,057.84 
2,117.41 
209.84 
1,025.90 

1,191.77 
1,056.96 
2,110.81 
209.84 
1,020.12 

1,191.77 
1,063.20 
2,104.87 
209.84 
1,020.12 

Total    .        .        .        .        . 

MIDDLE  ATLANTIC 

New  York       ..... 

8,180.85 

8,167.21 

8,212.12 

New  Jersey     
Pennsylvania  . 
Delaware         ..... 
Maryland         ..... 
District  of  Columbia 

2,242.56 
10,784.54 
333.63 

1,368.98 
24.70 

2,266.64 
10,991.97 
334.86 
1,364.45 
24.70 

2,269.61 
11,161.45 
333.60 
1,406.81 
24.70 

Total    

CENTRAL  NORTHERN 
Ohio        
Michigan         
Indiana  
Illinois    ...... 
Wisconsin       ..... 

9,023.61 
8,459.65 
6,834.75 
11,502.38 
6,921.40 

9,163.97 
8,467.76 
6,863.03 
11,742.10 
7,014.78 

9,243.26 
8,521.46 
7,046.90 
11,959.09 

7,188.18 

Total    

166 


ADVANCED  BOOK  OF  ARITHMETIC 


GROUP  AND  STATE 

19O3 

19O4 

1905 

SOUTH  ATLANTIC 

Virginia           

3,833.09 

3,823.67 

3,862.11 

West  Virginia          .... 

2,565.49 

2,820.82 

2,966.05 

North  Carolina       .... 

3,790.73 

3,913.86 

4,015.58 

South  Carolina       .... 

3,112.48 

3,146.24 

3,184.19 

Georgia  

6,109.21 

6,298.97 

6,516.61 

Florida   

3,469.92 

3,585.83 

3,635.38 

Total    

GULF  AND  MISSISSIPPI  VALLEY 

Alabama         ..... 

4,442.69 

4,590.89 

4,758.57 

Mississippi      ..... 

3,156.56 

3,367.23 

3,541.04 

Tennessee       ..... 

3,355.19 

3,484.92 

3,606.88 

Kentucky         ..... 

3,193.31 

3,261.56 

3,355.07 

Louisiana        

3,419.38 

3,592.68 

3,764.17 

Total    

SOUTHWESTERN 

Missouri  

7,316.62 

7,797.18 

7,859.57 

Arkansas         ..... 

3,651.28 

3,946.54 

4,165.72 

Texas      .         .         .         . 

11,308.05 

11,614.13 

11,949.02 

Kansas    ...... 

8,810.50 

8,841.09 

8,874.58 

Colorado          

4,852.44 

4,989.85 

5,093.20 

New  Mexico    

2,450.02 

2,441.93 

2,596.64 

Indian  Country       .... 

2,320.02 

2,585.69 

2,686.47 

Oklahoma        ..... 

2,359.52 

2,635.64 

2,836.19 

Total     

Find  the  total  mileage    for   each   group    of    states   as 
indicated. 


Add: 

891     Q3     ^5 
'     %>  "*¥'  °6* 


9. 

10.  3i,2|,5f2. 

11.  54,  9A,  2A. 


12.  If,  51  21. 

13.  7f,6&,»&. 

14.  9&,5|,  7|. 
is.  12|,  llf  ,  61 


ADDITION  167 

16.  2°  IV  50",  7°  24'  30",  9°  27'  37",  128°  14'  43". 

17.  2  ft.  9  in.,  7  ft.  3  in.,  9  ft.  11  in.,  15  ft.  7  in. 

18.  8  qt.  1  pt.,  9  qt.  1  pt.,  15  qt.,  12  qt.  1  pt. 

19.  4  gal.  2  qt.,  7  gal.  3  qt.,  9  gal.  1  qt.,  8  gal.  3  qt. 

20.  5  pk.  7  qt.,  9  pk.  3  qt.,  12  pk.  5  qt.,  13  pk.  4  qt. 

21.  3  bu.  3*pk.,  9  bu.  2  pk.,  7  bu.  1  pk.,  4  bu.  3  pk. 

22.  12  hr.  15  min.,  15  hr.  8  min.,  17  hr.  42  min.,  5  hr. 
13  min. 

23.  5  da.  12  hr.,  18  da.  17  hr.,  13  da.  18  hr.,  5  da.  3  hr. 

24.  15  yd.  2  ft.,  25  yd.  1  ft.,  32  yd.  2  ft.,  9  yd.  1  ft. 

25.  How  many  times  does  a  clock  strike  in  24  hours? 

26.  If  4  jars  contain  3.92  liters,  7.84  liters,  9.57  liters, 
and  6.3  liters  respectively,  how  many  liters  are  in  the  four 
jars? 

27.  The  dimensions  of  a  table  are  8  ft.  3  in.  by  3  ft. 
7  in.     How  many  feet  are  in  its  perimeter? 

28.  The  Galveston  Sea  Wall  was  constructed  by  Gal- 
veston  County  and  the  United  States  Government;  the 
former  built  3.5  miles,  and  the  latter  .87  mile.      Find  the 
total  length  of  the  sea  wall. 

In  its  construction  there  were  used  1150  carloads  of 
cement,  6100  carloads  of  crushed  rock,  1400  carloads  of 
round  piling,  475  carloads  of  sheet  piling,  4300  carloads 
of  riprap,  and  6  carloads  of  reenforcing  rods.  How  many 
carloads  of  material  were  used  in  its  construction  ? 

How  many  miles  would  the  cars  extend  if  placed  end 
to  end,  allowing  39.6  ft.  to  a  car? 

29.  The  decapod  locomotives  operating  between  Clarion 
Junction  and   Freeman,    Ohio,  weigh   268,000  Ib.    each. 
Express  this  weight  in  tons. 


168  ADVANCED  BOOK  OF  ARITHMETIC 


SUBTRACTION 

Subtraction  is  the  inverse  of  addition. 
To  subtract  7  from  16  is  to  find  a  number  which  added 
to  7  will  make  16. 

Example  l.    Subtract  63  from  92.  % 

92 

no         PROCESS.    3  and  9  are  12  ;  write  9,  carry  1.    1  and 

on     6  are  7,  7  and  2  are  9  ;  write  2. 

Example  2.    Subtract  6.3  from  9.2. 
9.2 

6.3     The  process  is  the  same  as  in  Example  1. 
2^9 

Example  3.    Subtract  6  hr.  3  min.  from  9  hr.  2  min. 
HR.     MIN.        PROCESS.    3  min.  and  59  min.  make  1  hr.  and 
2      2  min.;    write  59  min.,  carry  1  hr.     1  hr.  and 

6  _  §     6  hr.  are  7  hr.     7  hr.  and  2  hr.  are  9  hr.  ;  write 
2       59     2hp- 

Example  4.    Subtract  6|  from  9|. 
92. 

PROCESS.     |  and  -|  are  1^  ;  write  -|,  carry  1.    1  and 

-~     6  are  7,  7  and  2  are  9  ;  write  2. 

Example  5.    From  75,218  take  the  sum  of  4799,  3928, 
9476,  8873. 

75218  PROCESS.  3,  9,  17,  26  ;  26  and  2  are  28.  Write 
4799  2,  carry  2.  2,  9,  16,  18,  27  ;  27  and  4  are  31.  Write 
3928  4,  carry  3.  3,  11,  15,  24,  31  ;  31  and  1  are  32. 
9476  Write  1,  carry  3.  3,  11,  20,  23,  27;  27  and  8 
8878  are  35.  Write  8,  carry  3.  3  and  4  are  7.  Write 

48142      4      The  remain(ier  is  48,142. 

This  example  shows  the  practical  value  of  this  method 

of  subtraction.     (Austrian  Method.) 


SUBTRACTION 


169 


EXERCISE  94 

l.    Exports  of  domestic  manufactures  from  the  United 
States  for  the  years  ending  June  30, 1897,  and  1907  : 

ARTICLE  1897  1907 

Iron  and  steel,  manufactures  of     ...  $57,497,872  $181,530,871 

Copper,  manufactures  of 31,621,125  88,791,225 

Wood,  manufactures  of 35,679,964  79,704,395 

Oils— mineral,  refined 56,463,185  78,228,819 

Leather  and  manufactures  of     ....  19,161,446  45,476,960 

Cotton,  manufactures  of 21,037,678  32,305,412 

Agricultural  implements 5,240,686  26,936,456 

Naval  stores 9,214,958  21,686,752 

Carriages,  cars,  and  other  vehicles    .     .  9,952,033  20,513,407 

Chemicals,  drugs,  dyes,  and  medicines   .  8,792,545  18,220,630 

Instruments  and  apparatus 3,054,453  14,661,455 

Paper  and  manufactures  of 3,333,163  9,856,733 

Paraffin  and  paraffin  wax       4,957,096  9,030,992 

Fibers,  manufactures  of    ......  2,216,184  3,308,112 

India  rubber,  manufactures  of  ....  1,926,585  7,428,714 

Furs  and  skins 3,284,349  7,139,221 

Books,  maps,  engravings,  etc.     ....  5,647,548  5,813,107 

Tobacco,  manufactures  of 5,025,817  5,735,613 

Brass  and  manufactures  of 1,171,431  4,580,455 

Gunpowder  and  other  explosives   .     .     .  1,555,318  4,082,402 

Paints,  pigments,  and  colors 944,536  3,391,988 

Soap 1,136,880  3,806,097 

Musical  instruments 1,276,717  3,252,063 

Nickel  and  manufactures  of 726,789  3,218,862 

Clocks,  watches 1,770,402  3,160,272 

Coke 547,046  3,013,088 

Glass  and  glassware 1,208,187  2,604,717 

All  other  articles 19,799,642  47,295,739 


Find  the  increase  in  the  exports  of  each  of  the  above 
articles,  or  group  of  articles,  and  verify  your  work. 


170  ADVANCED  BOOK  OF  ARITHMETIC 

Find  the  difference  between  : 

2.  200  and  .02.  10.    $403.05  and  192.89. 

3.  400  and  1.37.  11.    160.52  and  $23.87. 

4.  $75  and  73^.  12.    100  and  .01. 

5.  $700  and  $2.84.  13.    6.29  and  2.9924. 

6.  $100  and  $1.75.  14.    5.001  and  4.0073. 

7.  $1000  and  5^.  15.    7.2  and  2.77. 

8.  $324.80  and  $100.99.         16.    11  and  1.5. 

9.  $70.73  and  $19.94.  17.    17.3  and  11.9. 

18.  The  square  of  6.715  and  the  square  of  .285. 

19.  7f  and  41J.  25.    9T^  and  3J. 

20.  18f  and  7f .  26.    6T9g  and  3|. 

21.  9|  and  4|.  27.    10T\  and  1\. 

22.  21 1  and  11 U.  28.    19|  and  84. 

o  1  o  o  o 

23.  7T3T  and  2J.  29.    12^  and  9^-. 

24.  8T3g  and  5T%.  30.    23{£  and  12 Jf . 

31.  5  ft.  7  in.  and  4  ft.  9  in. 

32.  17  ft.  3  in.  and  12  ft.  8  in. 

33.  19  ft.  1  in.  and  9  ft.  4  in. 

34.  27  ft.  3  in.  and  18  ft.  4  in. 

35.  9  Ib.  2  oz.  and  4  Ib.  7  oz. 

36.  17  Ib.  6  oz.  and  5  Ib.  11  oz. 

37.  33  Ib.  2  oz.  and  18  Ib.  8  oz. 

38.  12  hr.  10  min.  and  9  hr.  24  min. 

39.  90°  and  34°  14'  15". 

40.  180°  and  115°  4'  50". 

41.  180°  and  the  sum  of  56°  16',  and  92°  18'. 

42.  15  pk.  3  qt.  and  3  pk.  7  qt. 


SUBTRACTION  171 

43.  23  pk.  5  qt.  and  13  pk.  6  qt. 

44.  From  40,000  take  the  sum   of   3211,   4711,    5283, 
9438. 

45.  From  50,580  take  the  sum  of  19,311,  12,218,  1273, 
5559. 

46.  From  18,900  take  the  sum  of  3419,  3428, 4584,  2293. 

47.  A  man  owns  two  houses  worth  respectively  $2390 
and  $4575 ;  he  has  deposited  in  the  bank  $3280  ;  he  owes 
two  notes  for  $783  and  $870.     How  much  is  he  worth  ? 

48.  The   area  of   the  British    Isles   is   120,975  square 
miles;  the  area  of  Texas  is  265,780  square   miles.     By 
how  many  square  miles  does  the  area  of  Texas  exceed  the 
area  of  the  British  Isles  ? 

49.  The  population  of  the  Chinese  Empire  is  433,553,000; 
of  the  British  Empire,  363,900,000 ;  of  the  Russian  Em- 
pire,  141,000,000;    of    the    United    States,   exclusive   of 
colonial  possessions,  84,150,000 ;  of  Germany,  60,478,000. 
How  many  more  people  are  in  the  United  States  than  in 
Germany  ?     In  the  British  Empire  than  in  Russia,  United 
States,  and  Germany  combined?     By  how  many  does  the 
population  of   China   exceed   the   population   of   Russia, 
United  States,  and  Germany  together? 

50.  The   areas   of   Maine,   New  Hampshire,  Vermont, 
Massachusetts,  Rhode  Island,  and  Connecticut  in  square 
mrles  are  respectively  :     33,040,  9305,  9565,  8315,  1250, 
4990.     The  area  of  California  is  158,360  square  miles. 
By  how  many  square  miles  does  the   area  of  California 
exceed  the  area  of  the  six  New  England  states? 

51.  In  going  from  Galveston  to  Chicago  by  rail,  a  dis- 
tance of  1410  miles,  a  man  travels  the  first  day  345  miles ; 
the  next  day,  201  miles ;  the  third  day,  290  miles.     How 
far  is  he  from  Chicago  at  the  end  of  the  third  day  ? 


172  ADVANCED  BOOK  OF  ARITHMETIC 

MULTIPLICATION 

If  one  factor  of  the  product  is  multiplied  by  a  number, 
and  the  other  factor  divided  by  the  same  number,  the  product 
will  be  unchanged. 

Thus,  84  x  20  =  1680. 

420  x  4  =  1680.  Here  84  is  multiplied  by  5,  and 
20  is  divided  by  5. 

Example  1.  Multiply  3782  by  234. 
3782  or    3782 

234  234 

15128  =  4  x  3782  7564  =  200  x  3782 
11346  =  30  x  3782  11346  =  30  x  3782 
7564  =  200  x  3782  15128  =  4  x  3782 


884988  =  234  x  3782      884988  =  234  x  3782 

To  multiply  integers,  write  multiplicand  and  multiplier  so 
that  units  of  the  same  name  stand  in  the  same  column,  then 
multiply  the  multiplicand  by  each  digit  of  the  multiplier, 
placing  the  first  figure  of  each  partial  product  directly  under 
the  digit  of  the  multiplier  producing  it,  and  add  the  partial 
products. 

.Example  2.    Multiply  17.32  by  .47. 
17.32         PROCESS.     The  numbers  are  multiplied  as  if 
.47     both  were  integers  ;  then  beginning  at  the  right 


12  124  of  the  product  four  places  are  pointed  off,  that  is 
69  28  the  number  of  decimal  places  in  multiplicand  and 
8.1404  multiplier  combined.  This  may  be  readily  seen 
by  multiplying  the  multiplier  by  100  and  dividing  the 
multiplicand  by  100. 

Compare  with  explanation  page  59. 


MULTIPLICATION  173 

Example  3.    Multiply   4J  by  2|. 

2|=| 

o  o 

13 

Therefore,  4|  x  2|  =  3$.  x  f  =  ^  ^|  =  13. 

EXPLANATION.  If  the  first  factor  is  multiplied  by  8, 
the  result  is  39,  and  if  the  second  factor  is  multiplied  by 
3,  the  result  is  8.  Hence, 

8  x  3  x  required  product  =  39  x  8. 

Therefore,  required  product  =  — 

o    X  o 

Compare  with  explanation  on  page  46. 
Example  4.    Multiply  5  gal.  2  qt.  1  pt.  by  9. 

PROCESS.     9  times  1  pt.  =  4  qt.  1  pt. ;  write 

GAL.      QT.     PT. 

5     2     1     *  pk*  carry  4  qt.     9  times  2  qt.  =  18  qt.     18 
9     qt.  +  4  qt.  =  22  qt.  =  5   gal.   2    qt. ;    write   2 
5Q 2     I     *$">  carry  5  gal.     9  times  5  gal.  =  45  gal.     45 
gal.  -f-  5  gal.  =  50  gal. 

PARTICULAR  SHORT  METHODS   OF  MULTIPLICATION 

5  =  1  of  10  75  =  100  -  \  of  100 

25  =  £  of  100  875  =  1000  -  £  of  1000 
125  =  |  of  1000  99  =  100-1 

.16|  =  |  97  =  100-3 

Example  1.    Multiply  97.3  by  125. 

125  x  97.3  =  \  of  1000  x  97.3  =  \  of  97300  =  12162.5 

Example  2.    Multiply  29.374  by  993. 

29374.        =1000x29.374 
205.618=       7x29.374 

29168.382=    993x29.374 


174 


ADVANCED  BOOK  OF  ARITHMETIC 


EXERCISE  95 

1.  Multiply  each  of  the  following  numbers  by  10 : 

234,  350.2,  25.07,  .127,  .0788,  1.003. 

2.  Multiply  the  following  numbers  by  100  : 

505,  67.5,  27.28,  5.347,  .07954,  .00392. 

3.  Multiply  the  following  numbers  by  1000  : 

728,  96.4,  12.87, 1.732,  .0139,  .00782. 

4.  Multiply  the  following  numbers  by  10,000  : 

318,  25.4,  19.96,  18.832,  27.796,  .012. 

5.  Find   in  the    shortest  possible  way   the    following 
products : 

O)  2780  x  99 ;  9218  x  998 ;  7215  x  999. 
(5)  2.79x25;  3.18x125;  243x875. 
O)  78  x  .16|;  90  x  .331;  297  x  9998. 

6.  Multiply  5280  by  5280 ;  1020  by  1020. 

7.  Multiply  7309  by  256 ;  9417  by  735. 

8.  The  estimated  production  and  value  of  the  following 
cereal  crops  as  given  in  the  Annual  Report  of  the  Depart- 
ment of  Agriculture  for  the  year  1906  are  as  follows : 


CEREALS 

YIELD  PER  ACRE 

VALUE  PER  BUSHEL 

Corn     

bushels 

30.3 

cents 

39.9 

Wheat  

15.5 

66.7 

Oats      

31.2 

31.7 

live  . 

16.7 

58.9 

Barley  

28.3 

41.5 

Buckwheat    

18.6 

59.6 

Find  the  value  of  the  yield  per  acre  of  each  of  these 
cereal  crops. 


MULTIPLICATION 


175 


9.  The  number  of  bales  of  cotton  produced  in  Texas 
in  the  season  1904-05  was  2,598,949,  and  in  1903-04, 
3,214,133.  Allowing  500  Ib.  to  a  bale,  how  many  more 
pounds  of  cotton  were  produced  in  the  latter  year  than 
in  the  former? 

10.    The  estimated  production  and  value  per  ton  of  the 
hay  crop  for  the  year  1906  are  as  follows  : 


STATE 

YIELD  PER  ACRE 

PRICE  PER  TON 

New  Hampshire           .... 

tons 

1  15 

$12  50 

Massachusetts              .... 

1  31 

17.00 

Connecticut    

1.17 

15.00 

New  York                

1  28 

12.10 

New  Jersey    

1.32 

15.95 

Pennsylvania           

1  30 

13.40 

.Maryland  

1  26 

13.50 

Virginia     

125 

15.50 

South  Carolina   

146 

1525 

Georgia      

1  65 

15.75 

Alabama    

1.95 

13.30 

Louisiana  

1.93 

11.50 

Tennessee  

1  51 

13.45 

Kentucky  

1.35 

13.25 

Illinois  

.98 

12.50 

1.70 

5.50 

Kansas  

1.28 

6.25 

Colorado    

2.50 

9.50 

Utah           

4.00 

7.50 

Idaho    

2.95 

8.00 

1.85 

11.25 

Find  the  value  of  the  yield  per  acre  in  each  of  the  above 
states. 


176  ADVANCED  BOOK  OF  ARITHMETIC 

11.  A  piece  of  coal  taken  from  the  mine  at  Coos  Bay, 
Oregon,  had  the  following  composition  by  weight: 

Moisture  =.1042 

Combustible  matter  =  .4221 
Fixed  carbon  =  .4318 

Ash  =  .0419 

Find  the  amount  of  each  in  87  tons  of  this  coal:  in  783 
tons.     Check  your  answers. 

12.  Find  to  the  nearest  cent  the  value  of  each  of  the 
following  articles: 

(a)  25J  bu.  corn  @  42|  ^  per  bu. 
(6)  12T^  bu.  wheat  @  69 J^  per  bu. 
0)  28|  bu.  oats  @  25J  per  bu. 
(d)  16f  bu.  rye  @  60|  $  per  bu. 
O)  201  bu.  barley  @  47f  ^  per  bu. 
(/)  4|  Ib.  wool  @  61  ^  per  Ib. 
(#)  497  Ib.  cotton  @  llf  $  per  Ib. 
(A)  512  Ib.  cotton  @  10{  f  per  Ib. 

13.  The  inside  dimensions  of  the  floor  of  a  box  car  are 
40  ft.  -|  in.  by  8  ft.  6  in.     Find  the  perimeter  of  the  floor. 

14.  The  inside  dimensions  of  the  floor  of  a  refrigerator 
car  are  28  ft.  9|  in.  by  8  ft.  1^  in.     Find  its  perimeter. 

15.  Multiply  5  yd.  2  ft.  by  8 ;  9  ft.  8  in.  by  7. 
Find  the  product  of : 

16.  9  Ib.  4  oz.  by  5 ;  16  Ib.  11  oz.  by  9. 

17.  3  hr.  20  min.  30  sec.  by  6 ;  7  hr.  17  min.  by  9. 

18.  53°  12'  by  10 ;  68°  12'  18"  by  5. 

19.  4  pk.  7  qt.  by  6;  7  pk.  3  qt.  by  8. 

20.  5  gal.  2  qt.  by  9;  9  gal.  3  qt.  by  12. 


DIVISION 


177 


DIVISION 

Division  is  the  inverse  of  multiplication. 

To  divide  84  by  7  means  to  find  a  number  which  multi- 
plied by  7  gives  84. 

If  the   divisor  and  dividend    are    both    multiplied    by  the 
same  number,  the  quotient  remains  unchanged. 
Thus,  96  -  8  =  12. 

(96  x  6)  -  (8  x  6)  =  12. 

Example  l.    Divide  2483  by  7. 

7)2483  PROCESS.     7  is  contained  in  24  hundreds  3 

354|-  hundred  times,  remainder  3  hundreds  ;  3  hun- 
dred =  30  tens.  30  tens  and  8  tens  =  38  tens.  7  is  con- 
tained in  38  tens  5  tens  times,  remainder  3  tens  ;  3  tens  = 
30  units.  30  units  and  3  units  =  33  units.  7  is  contained 
in  33  4  times,  with  a  remainder  of  5. 

Example  2.   Divide  .437  by  1.92. 

.2276+ 

PROCESS.     Move  the  decimal  point  two 

places  to  the  right  in  divisor  and  dividend  ; 
this  multiplies  both  by  100.  Then  write 
each  quotient  figure  directly  above  the 
right-hand  figure  of  the  partial  dividend 
which  produces  it.  Write  the  decimal 
point  in  the  quotient  above  the  decimal 
point  in  the  dividend. 


192^43  7 
QQ  4 


1  344 


Example  3. 


Divide  3|  by  6£. 
|  =(3|  x  3  x  2) 


(61  x  3  x  2)  = 


Multiply  divisor  and  dividend  by  3  x  2. 
divisor  and  dividend  both  whole  numbers. 


This  makes 


178 


ADVANCED   BOOK  OF  ARITHMETIC 


Example  4.    To  how  many  long  tons  are  3.30693  short 
tons  equivalent? 

100 
3.30693  x  nn  _  330.693  __ 


112 

To  check  an  example  in  division,  multiply  the  quotient 

by  the  divisor. 

EXERCISE  96 

l.    The  estimated  acreage,  production,  and  value  of  the 
potato  crop  by  states  for  the  year  1905  are  as  follows  : 


STATE 

ACREAGE 

PRODUCTION 

FARM  VALUE 

New  Hampshire     

acres 
19,700 

bushels 
2,367,000 

dollars 
1,704,000 

Rhode  Island    

6,490 

811,200 

722,000 

7,680 

714,000 

421,200 

25,900 

1,993,000 

1,355,000 

Florida     

4,110 

308,200 

369,900 

5,860 

644,900 

548,200 

34,400 

3,025,100 

1,754,500 

242,000 

16,203,000 

9,073,700 

149,000 

11,186,000 

7,494,600 

86,100 

7,059,000 

3,882,614 

25,400 

2,415,000 

917,800 

Nevada                         

2,800 

336,700 

276,100 

Find  the  number  of  bushels  yielded  per  acre  in  each 
state,  and  the  average  price  per  bushel  in  cents. 
Divide  correct  to  four  decimal  places: 

2.  128.016  by  420.  6.    .02734  by  .044. 

3.  2.3774  by  7.8.  7.    .035936  by  .0888. 

4.  10.4987  by  3.2.  8.    1.57899  by  .639. 

5.  .77087  by  .479.  9.    60.247  by  78.8. 

10.    5.0748  by  3.88. 


DIVISION  179 

Divide  : 

11.  21  by  1^T.  16.    93f  by  62f  . 

12.  48  by  2f  .  17.    17f  by  9fi. 

13.  42bylTV  18. 

14.  72  by  3f.  19. 

15.  72lby4f  20.    2ilby2ff. 
Find  the  value  of  : 


SOLUTION.     fxfx|xf= 

22.  £  X  If  -«-  2£  X  TV  24.     |  X  2f  X  1*  -«-  If. 

23.  |  X  If  -*-  1J  -H  8f  25.     1^-8J-!.3JXJ. 

26.  If  -I-  (I  -8-  41)   X  4f  . 

27.  |  Of  If  Of  31  -  I  Of  If 

HINT,     i  x  f  x  I  -i-  if. 

Observe  the  divisor  is  f  of  1-J. 

28.  |  of  If  of  2|  -  f  of  If. 

29.  11  of  If  of  TV^  A  off. 

30.  |of4|X29o-^f|. 

31.  If  Of  3f  Off-*-!  Off. 

Express  in  long  tons  : 

32.  3.36  T.,  6.6139  T.,  .00992  T.,  4.4092  T. 

Express  in  short  tons  : 

33.  1.9684  long  T.,  6.8894  long  T.,  .004921  long  T. 
Express  in  troy  pounds  : 

34.  5  Ib.  avoirdupois,  13.228  Ib.  avoirdupois,   6613.87 
Ib.  avoirdupois. 


180  ADVANCED  BOOK  OF  ARITHMETIC 

Express  in  avoirdupois  pounds  : 

35.  7  Ib.  troy,  13.396  Ib.  troy,  5.358  Ib.  troy. 

36.  Express  1  ft.  as  a  decimal  of  1  mi. 

37.  Express  1  rd.  as  a  decimal  of  1  mi. 

38.  Express  1  sq.  rd.  as  a  decimal  of  1  A. 

39.  Express  1  A.  as  a  decimal  of  1  sq.  mi. 

40.  Express  1  sq.  yd.  as  a  decimal  of  1  A. 

41.  Express  1  Ib.  as  a  decimal  of  1  ton. 

42.  A  lot  is  40  by  120  feet.     How  many  such  lots  make 
40  acres? 

43.  How  many  barrels  of  31 J  gallons  each  will  a  rec- 
tangular tank  12  ft.  by  8  ft.  and  5  ft.  deep  hold  ?    (Allow 
7^  gal.  to  a  cubic  foot.) 

44.  The  weight  of  a  half  dollar  is  12J  grams.     How 
many  half  dollars  can  be  made  out  of  7500  grams  of  stand- 
ard silver  ? 

45.  Find  the  cost  of  boring  an  artesian  well  1400  feet 
deep  at  $4  a  linear  foot  for  the  first  900  feet,  $4.50  per 
linear  foot  for  the  next  200  feet,  $  5  per  linear  foot  for  the 
next  100  feet,  $  5.50  per  linear  foot  for  the  next  100  feet, 
and  $  6  per  linear  foot  for  the  remainder. 

46.  The  rails  of  the  Great  Western.  Railway,  England, 
weigh  97|  Ib.  per  yard.     Find  in  tons  the  weight  of  the 
rails  required  to  construct  1  mile  of  this  railway. 

47.  Express  in  the  ordinary  decimal  notation  : 

3.27  x  106,  17.45  x  109,  9.4  x  103,  7.3  x  106. 

48.  Given   10-i  =  TV    10-2  =  _i_,    10-3  =  _i_ 

=  TWO  0>  10~5  =  TFoVoT'  1°~6  =  TOOOOOO- 

Express  in  the  ordinary  decimal  notation  : 
(a)  3.2  x  10-3,  4.71  x  10~6,  9.83  x  10'3. 
(6)   4.98  x  10-5,  9.371  x  10"6,  4.329  x  10-4. 


LONGITUDE  AND  TIME  181 

LONGITUDE  AND  TIME 

A  meridian  is  an  imaginary  line  running  due  north  and 
south  from  pole  to  pole. 

Longitude  is  the  distance,  expressed  in  circular  arc 
measure,  east  or  west  from  the  prime  or  standard  me- 
ridian. 

The  meridian  through  any  particular  place  may  be  used 
as  the  prime  meridian.  The  meridians  through  the  ob- 
servatories of  Greenwich,  Washington,  Paris,  Madrid, 
Rome,  Stockholm,  Pulkova,  and  Lisbon  have  been  used 
as  prime  meridians  by  the  nations  to  which  these  cities 
belong.  The  International  Geodetic  Congress,  which  met 
at  Washington  in  1884,  recommended  that  the  meridian 
passing  through  the  observatory  at  Greenwich,  a  suburb 
of  London,  be  the  prime  meridian.  This  recommendation 
is  now  generally  adopted  by  the  great  nations  of  the  world. 
The  meridian  of  Greenwich  is  taken  as  prime  meridian  in 
this  book. 

Longitude  is  reckoned  in  either  direction  halfway  around 
the  earth  from  the  prime  meridian.  The  greatest  longi- 
tude a  place  can  have  is  180°  E.  or  180°  W.  The  meridian 
180°  E.  or  180°  W.  of  the  prime  meridian  is  a  continuation 
of  the  prime  meridian  on  the  other  side  of  the  earth,  and 
forms  with  the  prime  meridian  what  is  called  a  great  circle 
passing  through  the  poles. 

The  earth  rotates  on  its  axis  from  west  to  east.  Con- 
sider two  places  not  on  the  same  meridian ;  for  example, 
New  York  City  and  St.  Louis.  New  York  being  farther 
east  will  come  the  sooner  under  the  influence  of  the  sun's 
rays.  Therefore,  when  it  is  noon  in  New  York  City  it  is 
before  noon  in  St.  Louis.  Since  the  earth's  motion  is 
uniform,  and  furthermore,  since 


182  ADVANCED  BOOK  OF  .ARITHMETIC 

in  24  hr.  the  earth  rotates  360°, 
/.in  1  hr.  the  earth  rotates  15°; 
.'.in  1  min.  the  earth  rotates  15' ; 
/.in  1  sec.  the  earth  rotates  15". 

A  difference  of  15°  of  longitude  corresponds  to  a  differ- 
ence of  1  hr.  of  time.  A  difference  of  15'  of  longitude 
corresponds  to  1  min.  of  time.  A  difference  of  15"  of 
longitude  corresponds  to  1  sec.  of  time. 

Hence,  to  convert  difference  of  longitude  into  difference 
of  time,  divide  by  15. 

EXERCISE  97 

1.  When  it  is  noon  at  London,  what  is  the  time  at  New 
Orleans,  90°  W.  ? 

2.  When  it  is  9  o'clock  A.M.  on  the  meridian  75°  W., 
what  is  the  time  on  the  meridian  90°  W.  ? 

3.  The   longitude  of  Denver  is  105°  W.      When  it  is 
3  o'clock  P.M.  in  Denver,  what  is  the  time  in  London? 

4.  Two  places  differ  in  longitude  by    20°.      What  is 
their  difference  in  time  ? 

5.  A  person  travels  east  15°.      What  change  must  he 
make  in  the  time  indicated  by  his  watch  so  that  it  may 
indicate  local  time  ?     Supposing  he  goes  the  same  distance 
west,  what  change  must  be  made  in  the  time  indicated  by 
his  watch  ? 

6.  When  it  is  noon,  in  London,  what  is  the  longitude 
of  the  places  in  which  it  is  4  o'clock  P.M.  ?    5  o'clock  A.M.? 

7.  When  it  is  2  o'clock  P.M.   in  Washington,  what  is 
the  time  in  places  30°  W.  of  Washington  ?  in  places  75°  E. 
of  Washington? 

8.  What  is  the  difference  in  longitude  between  places 
which  differ  in  time  by  2  hr.  30  min.  ?  by  4  hr.  10  min.  ? 


LONGITUDE  AND  TIME  183 

9.  If  a  person  travels  from  Denver  to  New  York,  will 
his  watch  be  fast  or  slow  when  he  reaches  New  York,  and 
how  much  ? 

10.  To  how  many  hours  "does  a  difference  of  80°   in 
longitude  correspond? 

11.  What  difference  in  longitude  corresponds  to  a  dif- 
ference of  4  hr.  20  min.  in  time  ? 

12.  A  person  living  on  the  90th  meridian  W.  wishes 
to  send  a  telegram  to  a  bank  in  New  York  City,  directing 
the  bank  to  pay  on  the  same  day  a  sum  of  money.     Up  to 
what  hour  in  the  afternoon  may  he  do  this,  allowing  30 
minutes  for  the  transmission  of  the  telegram,  taking  the 
longitude  of  New  York  as  75°  W.  ?     (New  York  banks 
close  at  3  P.M.) 

13.  At  places  on  the  same  parallel  of  latitude  the  sun 
rises  at  the  same  instant  local  time.     How  many  minutes 
earlier  does  the  sun  appear  to  a  person  who  travels  1°E.? 

LONGITUDES  OF  CITIES  REFERRED  TO  IN  THIS  CHAPTER 


Austin, 
Baltimore, 
Bangor, 
Bismarck, 

97° 
76° 
68° 
100° 

44'  W. 

37'  W. 
47'  W. 
47'  W. 

Galveston, 
Havana, 
Honolulu, 
Louisville, 

94°  47' 
82°  21' 
157°  52' 
85°  46' 

W. 
30" 
W. 

W. 

W. 

Boston, 

71° 

3' 

50" 

W. 

Melbourne, 

144° 

58' 

32"  E. 

Brisbane, 

153° 

2' 

E. 

Manila, 

120° 

58' 

3 

n 

E. 

Buenos  Ayres, 

58° 

22' 

14" 

W. 

Mexico  City, 

99° 

6' 

39 

n 

W. 

Charleston, 

79° 

52'  58" 

W. 

Montreal, 

73°  33' 

4 

n 

W. 

Chicago, 

87° 

40' 

W. 

New  Orleans, 

90° 

3' 

28 

11 

W. 

Cincinnati, 

84° 

24' 

W. 

New  York 

> 

74° 

0' 

24 

n 

W. 

Constantinople, 

29° 

0' 

50" 

E. 

Norfolk, 

76° 

17' 

22 

n 

W. 

Detroit, 

83° 

3' 

W. 

Paris, 

2° 

20' 

15 

n 

E. 

Dublin, 

6° 

20' 

30" 

W. 

Pekin, 

116° 

29' 

E. 

184  ADVANCED  BOOK  OF  ^ARITHMETIC 

Pensacola,  87°  16'    6"  W.  St.  Petersburg,    30°  19' 40"  E. 

Philadelphia,  75°    9'    3"W.  San  Francisco,  122°  24'  32"  W. 

Portland,  122°  40' W.          Savannah,  81°    5'25"W. 

Providence,  71°  24' 20"  W.  Tientsin,  117°  11' 44"  E. 

Borne,  12°  28'  40"  E.    Tokyo,  139°  44'  30"  E. 

St.  Louis,  90°  16'  W.         Washington,        77°    0'  36"  W. 

In  recent  years  scientific  publications  often  give  longi- 
tudes in  terms  of  time,  the  -f  sign  denoting  west  and  the 
—  sign  denoting  east. 

H.     M.      S.  H.     M.      S. 

Harrisburg,        +5      7    32  Adelaide,        -  9    14      2 

Milwaukee,        +  5    51     37  Omaha,  -f-  6    23    46 

Example  1.  Find  the  difference  between  the  longi- 
tudes of  Austin  and  Honolulu. 

SOLUTION.     Honolulu,  157°  52'  W. 
Austin,        97°  44'  W. 

60°     8' 
.-.  Honolulu  is  60°  8'  farther  west  than  Austin. 

Example  2.  Find  the  difference  between  the  longi- 
tudes of  Galveston  and  Constantinople. 

SOLUTION.     Galveston,          94°  47'  W. 

Constantinople,  29°    0'  50"  E. 

Here,  the  places  are  on  opposite  sides  of  the  prime 
meridian.  By  going  east  from  Galveston  94°  47',  one 
arrives  at  the  prime  meridian,  and  by  going  29°  0'  50" 
still  farther  east,  he  arrives  at  the  meridian  of  Con- 
stantinople. Hence,  the  difference  between  the  longitudes 
is  (94°  47'  +  29°  0'  50")  =  123°  47'  50". 

To  find  the  difference  in  the  longitudes  of  two  places  : 
(l)  Subtract  their  longitudes,  if  the  places  are  on  the  same 
side  of  the  prime  meridian.  (2)  Add  their  longitudes,  if 
the  places  are  on  opposite  sides  of  the  prime  meridian. 


LONGITUDE  AND  TIME  185 

EXERCISE  98 

Find  the  difference  in  longitude  between  : 

1.  Baltimore  and  Bismarck. 

2.  Bangor  and  Detroit. 

3.  Boston  and  Havana. 

4.  Buenos  Ayres  and  Chicago. 

5.  Charleston  and  Constantinople. 

6.  Cincinnati  and  Honolulu. 

7.  Cincinnati  and  Melbourne. 

8.  Havana  and  Rome. 

9.  Louisville  and  St.  Petersburg. 

10.  Constantinople  and  Tientsin. 

11.  Paris  and  Pekin. 

12.  Norfolk  and  Paris. 

13.  Montreal  and  Mexico  City. 

14.  Pensacola  and  Portland. 

15.  St.  Louis  and  St.  Petersburg. 

16.  Savannah  and  Dublin. 

17.  San  Francisco  and  Dublin. 

Example  1.    Find  the  difference  in  local  time  between 
Boston  and  Portland,  Ore. 

Portland     122°  40'             W. 

Boston          71°  3'  50"  W. 

15)50°  36'  10" 

3  22  25       Ans.    3  hr.  22  min.  25  sec. 

Example  2.    Find  the  difference  in  local  time  between 
Washington  and  Manila. 

Washington  77°     0'     36"  W. 
Manila          120°   58"      3"  E. 
15)197°   58'     39" 

13     11      54.6  Ans.  13  hr.  11  min.  54.6  sec. 


186  ADVANCED  BOOK  OF  ARITHMETIC 

EXERCISE   99 

Find  the  difference  in  the  local  time  of  : 

1.  Mexico  City  and  Montreal. 

2.  Philadelphia  and  San  Francisco. 

3.  Philadelphia  and  Dublin. 

4.  Norfolk  and  Tientsin. 

5.  Chicago  and  Tokyo. 

6.  St.  Louis  and  Rome. 

7.  Austin  and  St.  Petersburg. 

8.  Savannah  and  Paris. 

9.  Washington  and  Brisbane. 

10.  Cincinnati  and  Manila. 

11.  Havana  and  Louisville. 

12.  Rome  and  Manila. 

13..  New  Orleans  and  Portland. 

14.  Providence  and  St.  Petersburg. 

15.  Montreal  and  Tokyo. 

16.  Bangor  and  Melbourne. 

17.  Baltimore  and  Buenos  Ayres. 

Example  l.  When  it  is  noon,  February  22,  in  St.  Louis, 
it  is  15  min.  6  sec.  past  three  o'clock  A.M.,  Feb.  23,  in 
Adelaide,  Australia.  Find  the  longitude  of  Adelaide. 

SOLUTIOIST.  The  time  difference  between  St.  Louis  and 
Adelaide  is 

15  hr.  15  min.  6  sec. 

Multiply  by  15,  15 

228°       46'        30' ' 

.-.  Adelaide  is  228°  46'  30"  E.  of  St.  Louis.  Longitude 
of  St.  Louis  is  90°  16'  W.  .-.  longitude  of  Adelaide  = 
(228°  46'  30"  -  90°  16")  E.  =  138°  30'  30"  E. 


LONGITUDE  AND  TIME 


187 


EXERCISE   100 

Calculate  the  longitude  of  each  of  the  following  cities, 
the  time  difference  between  New  York  City  and  each  of 
them  being  given  : 

1.  Berlin,  5  hr.  49.5  min. 

2.  Brussels,  5  hr.  13.4  min. 

3.  Calcutta,  10  hr.  49.2  min. 

4.  Edinburgh,  4  hr.  43.2  min. 

5.  Hamburg,  5  hr.  35.8  min. 

6.  London,  4  hr.  55.9  min. 

7.  Madrid,  4  hr.  41.1  min, 

8.  Vienna,  6  hr.  1.2  min. 

9.  The  time  difference  between  London  and  Amherst, 
Mass.,  is  4  hr.   50  min.  3   sec.     Find  the  longitude   of 
Amherst. 

10.  Find  the  difference  in  the  time  of  sunrise  between 
two  points  in  the  same  latitude  and  which  differ  in  longi- 
tude by  39°  20'. 


oJfe^ncT/^  B«*l3Sfc* 


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188  ADVANCED  BOOK  OF  ARITHMETIC 

STANDARD   TIME 

Standard  time  is  the  time  of  a  fixed  meridian,  generally 
a  multiple  of  15°.  It  was  established  in  the  United  States 
in  1883  primarily  for  the  convenience  of  railroads.  It  is 
now  adopted  generally  throughout  the  civilized  world. 


STANDARD  MERIDIANS  AND  PLACES  USING  THEM 

0°.    Great  Britain,  Spain,  Belgium,  Holland. 
15°  E.    Germany,  Austria,  Italy,  Denmark,  Norway. 
30°  E.    South  Africa,  Egypt,  Turkey. 
821°  E.    British  India  (since  July  1,  1905). 
971°  E.    Burma  (since  July  1,  1905). 
120°  E.    West  Australia,  eastern  coast  of  China,  Phil- 
ippine Islands. 
135°  E.    Japan. 
142J°  E.    South  Australia. 

150°  E.    Victoria,  Queensland,  New  South  Wales. 
172|°  E.    New  Zealand. 

60°  W.    Newfoundland  and  Eastern  Canada. 
75°  W.    Eastern  belt  of  the  United  States. 
90°  W.    Central  belt  of  the  United  States. 
105°  W.    Mountain  belt  of  the  United  States. 
120°  W.    Pacific  belt  of  the  United  States. 
135°  W.    Alaska. 
150°  W.    Tahiti. 
1571°  W.    Hawaiian  Islands. 
France  uses  Paris  time,  Ireland  uses  Dublin  time. 


STANDARD  TIME  189 

EXERCISE  101 

1.  Mariners  carry  on  board  ships  chronometers  which 
keep  Greenwich  time.     When  it  is  noon,  local  time,  the 
chronometer  indicates  4  hr.  48  min.  P.M.     What  is  the 
longitude  of  the  ship  ? 

2.  When  it  is  10  o'clock  P.M.,  March  2,  in  Washing- 
ton, what  is  the  standard  time  in  Manilla?  Melbourne? 
Berlin? 

3.  When  it  is  2  o'clock  A.M.,  standard  time,  in  Den- 
ver, what  is  the  standard  time  of  London?  Manchester? 
Glasgow?  Tientsin?  Constantinople? 

4.  A  telegram  is  sent  from  Madrid  to  Washington  at 
9  o'clock  A.M.     Allowing  1  hr.  for  transmission,    when 
will  it  reach  Washington? 

5.  At  noon,  local  time,  a  chronometer  indicates  11 
o'clock  P.M.     What  is  the  longitude? 

6.  A  telegram  is  sent  from  Galveston  to  London  at  10 
o'clock  P.M.     When   will  it  be  received,  allowing  2  hr. 
for  transmission? 

7.  When  it  is  2  o'clock  A.M.  in  Washington,  standard 
time,  what  is  the  time  in  New  Zealand?  Tahiti?  British 
India? 

8.  The  San  Francisco  earthquake  occurred  April  18, 
1906,  at  5  A.M.     When    should  the   news  have  reached 
London?  Berlin?  Tokyo?  Adelaide?  (allowing  1  hour  for 
transmission). 

9.  When  it  is  noon  in  Paris,  France,  what  is  the  time  in 
Denver?  Natal?  Calcutta?  Wellington  (New  Zealand)? 

10.  When  it  is  9  o'clock  A.M.  in  Madras,  what  is  the 
time  in  St.  John's,  Newfoundland?  Chicago?  Sitka? 

11.  When  it  is  noon  in  the  Hawaiian  Islands,  what  is 
the  time  in  Cairo  (Egypt)  ?  Perth  (Western  Australia)  ? 


190  ADVANCED   BOOK  OF  ARITHMETIC 

APPROXIMATIONS.      CONTRACTED   PROCESSES. 
GENERAL  METHODS  OF   SOLUTION 

In  business  problems  results  of  computation  are  gen- 
erally required  to  be  correct  to  not  more  than  two  decimal 
places.  For  example  :  The  interest  on  $79.50  for  4  months 
at  7%  is  $1.855.  From  a  business  point  of  view  the 
answer  is  $1.85. 

In  all  practical  measurements  of  length  it  requires  skill 
and  long  practice  to  get  results  correct  to  more  than  three 
figures.  For  example :  A  surveyor  measures  the  length 
of  a  field  and  finds  it  to  be  3729  feet.  It  is  extremely 
probable  that  the  last  figure  in  this  result  is  not  correct. 
As  the  results  of  measurement  are  correct  to  only  three 
or  four  figures,  hence  it  is  useless  in  computation  to  give 
results  to  more  than  three  decimal  places. 

Before  undertaking  to  show  how  results  may  be  obtained 
correct  to  any  given  number  of  figures,  it  is  well  to  fix  in 
mind  the  following  facts : 

Tenths  multiplied  by  tenths  give  hundredtks. 

Tenths  multiplied  by  hundredths  give  thousandths. 

Tenths  multiplied  by  thousandths  give  ten-thousandths. 

Hundredths  multiplied  by  hundredths  give  ten-thousandths. 

Example  1.    Multiply  .0537928  by  43.27. 

Move  the  decimal  point  one  place  to  the 
.537928  left  in  the  multiplier. 

4.327  Move  the  decimal  point  one  place  to  the 

2.151712  right  in  the  multiplicand. 

1613784  These  changes  make  no  change  in  the 

1075856       product. 

3765496          Suppose  it  be  required  to  get  the  product 
2.327614456     correct  to  two  decimal  figures.     The  answer 
would  be  2.33. 


APPROXIMATIONS  191 

Write  the  units'  figure  of  the  multi- 

CONTRACTED  PROCESS  piier  under  the  third  decimal  figure  of 

.537982  the   multiplicand.      Multiply   4    by    7 

4.327         and  carry  4  from  4  multiplied  by  9  be- 

2.152  cause  36  is  nearer  40  than  30.     Multi- 

161  ply  the  remaining  figures  to  the  left  by 

11  4  in  the  usual  manner. 

4  As  tenths  multiplied  by  thousandths 

2.328  give  ten-thousandths,  multiply  3  by  3 

2.33  to  the  left  of  7  and  carry  2  from  3  times 

7.  3  times  5  are  15  and  1  make  16. 
As  hundredths  multiplied  by  hundredths  give  ten-thou- 
sandths, multiply  2  by  5  and  carry  1  from  2  times  3.  As 
thousandths  multiplied  by  tenths  give  ten-thousandths,  7 
is  multiplied  by  no  figure  of  the  multiplicand,  4  is  carried 
from  7  times  5. 

Compare  the  two  processes. 

Example  2.  Multiply  253.7  by  .079  correct  to  two  deci- 
mal figures. 

Move  the  decimal  point  in  the  multi- 

2.537  plier  two  places  to  the  right.     Move  the 

7.9         decimal  point  in  the  multiplicand   two 

17.759  places  to  the  left.     These  changes  make 

2.283  no  change  in  the  product.     Multiply  by 

20.04  7  in  the  usual  manner.     Multiply  by  9 

beginning  with  9  times  3,  and  adding  6 

to  the  product  which  is  the  figure  carried  from  9  times  7. 

Move  the  decimal  point  in  the  multiplier  so  that  it  con- 
tains one  integral  figure.  Move  the  decimal  point  in  the 
multiplicand  the  same  number  of  places  in  an  opposite  direc- 
tion. Place  the  units'  figure  of  the  multiplier  under  the  third 
place  of  the  multiplicand,  if  a  product  to  two  decimal  figures 


192  ADVANCED  BOOK  OF  ARITHMETIC 

is  required.  If  a  product  to  three  decimal  places  is  required, 
place  the  units'  figure  of  the  multiplier  under  the  fourth  deci- 
mal place  of  the  multiplicand.  Then  multiply  as  indicated 
in  the  above  examples. 

Example  3.    Multiply  .732  by  .864  correct  to  two  deci- 
mal figures. 

Begin  multiplying  by  8  by  taking 
.0732  the  product  8x3,  carrying  2  from  8     .0732 

8.64      x  2.     Begin  the  multiplication  by      468 

.586  6  with  6x7,  carrying  2  from  6  x       586 

44  3.     Begin  the  multiplication   by  4        44 

3  with  4x0,  carrying  3  from  4x7.       3 

.63  The  arrangement  in  the  right  mar-     .63 

gin  conserves  energy,  for  the  multi- 
plication by  each  figure  of  the  multiplier  is  begun  with 
the  figure  directly  above  it. 

Example  4.    Divide  120.005  by  17.293  correct  to  three 
decimal  figures. 

6.939+ 

17293)120005 
103758 
162470 
155637 


68330 
51879 
164510 
155637 
8873 

The  answer  correct  to  three  decimal  figures  is  6.940  as 
the  next  will  be  5. 


APPROXIMATIONS  193 

CONTRACTED  PROCESS          jn  this  example  the  quotient  is 

6.939+  required  correct   to  four   figures. 

17293)120005  The  divisor  contains  five  figures. 

103758  Whenever  the  divisor  contains  one 

16247  or  more  figures  than  are  required  in 

15564  the  quotient,  a  figure  may  be  struck 

683  off  the  divisor  in  place  of  annexing 

519  or  taking  down  a  figure,  as  is  usu- 

164  ally  done  in  getting  each  figure  of 

155  the  quotient. 

9  Compare  the  two  processes. 

GENERAL  METHOD 

Example  a.  If  3  acres  of  land  are  worth  $  129,  how 
much  are  5  acres  worth  at  the  same  rate  per  acre  ? 

Example  b.    If  8  masons  build  a  wall  in  18  days,  how 
long  would  it  take  9  masons  to  build  the  wall  ? 
Example  (a)  The  cost  of  3  acres  =  $  129. 

The  cost  of  1  acre   =  $  of  $  129. 
The  cost  of  5  acres  =  f  of  9 129  =  f  205. 
Example  (5)   The  time  8  masons  take  =  18  da. 

The  time  1  mason  takes  =  8  x  18  da. 
The  time  9  masons  take  =  |  x  18  da.  =  16  da. 
The  answer  in  Example  (#)  is  a  fraction  of  $129. 
The  answer  in  Example  (6)  is  a  fraction  of  18  days. 
The  solution  of  examples  of  this  character  consists  in 
multiplying  the  quantity  of  the  same  kind  as  the  answer 
by  a  fraction. 

If  the  answer  is  to  be  greater  than  the  given  quantity, 
form  the  fraction  so  that  the  numerator  is  greater  than  the 
denominator.  If  the  answer  is  to  be  less  than  the  given 
quantity,  form  the  fraction  so  that  the  numerator  is  less  than 
the  denominator. 


194  ADVANCED  BOOK  OF  ARITHMETIC 

Example  l.  If  a  dealer  sells  a  piano  for  $425,  thereby 
losing  15  %,  what  should  he  have  sold  it  for  to  make  a 
profit  of  15  %  ? 

In  this  example  85  %  of  cost  is  given,  and  115  %  of  cost 
is  sought.  The  answer  will  be  obviously  more  than 
$425. 

Hence,  iff-  x  $425  =  $  575,  Ans. 

Example  2.  A  kilometer  is  very  nearly  equivalent  to 
^  of  a  mile.  Express  a  mile  in  kilometers. 

5  eighths  of  1  mile  is  given  and  8  eighths  is  sought. 
Hence,  f  of  1  kilometer  =  1.6  kilometers. 

EXERCISE   102 

Solve  by  the  above  method. 

1.  If  6  horses  plow  a  field  in  9  days,  how  long  will  it 
take  9  horses  to  plow  the  same  field  ? 

2.  If  a  train  runs  in  3|  hours  between  two  stations  at 
the  rate  of  18  miles  an  hour,  how  long  will  it  take  a  train 
whose  speed  is  30  miles  an  hour  to  make  the  same  run? 

3.  If  5  acres  of  land  sell  for  $  423,  at  this  rate  what 
will  be  the  selling  price  of  7  acres  ? 

4.  If  22  yd.  of  cloth  are  bought  for  a  sum  of  money, 
how  many  yards  may  be  bought  for  the  same  sum  when 
the  price  falls  12  %  ? 

5.  Eight  horses  consume  a  quantity  of  corn  in  24  days. 
How  long  should  the  same  quantity  of  corn  last  12  horses? 

6.  The  minute  hand  of  a  clock   goes  360°  in  1  hour. 
How  many  degrees  does  it  go  in  22  minutes  ? 

7.  An  arc  of  75°  is  4  ft.  6  in.     How  many  feet  are  in 
the  circumference  of  the  circle  ? 


APPROXIMATIONS  195 

8.  If  2^  of  the  number  of  miles  from  Paris  to  Turin  is 
27|,  what  is  the  entire  distance  separating  the  cities? 

9.  If  ^  of  the  number  of  miles  from  New  York  City 
to  Panama  is  1727,  how  far  is  Panama  from  New  York? 

10.  Given  .9  of  the  distance  from  London  to  Constan- 
tinople as  1827  mi.,  how  many  miles  is  it  from  the  former 
to  the  latter? 

11.  If  |~|  of  the  distance  from  Hamburg  to  Vienna  is 
143  mi.,  find  the  distance  between  these  cities. 

12.  In  the  year  1902,  l^  of  the  United  States  internal 
revenue  receipts  from  tobacco  amounted  to  $22,852,687. 
Find  the  total  internal  revenue  receipts  from  tobacco  for 
that  year. 

13.  In  the  year  1902,  ^  of  the  excise  tax  in  the  United 
States  on  gross  receipts  under  the  War  Revenue  Law  of 
1898  amounted  to  $117,221.     Find  the  total  tax  on  gross 
receipts  in  1902. 

14.  In  the  year  1902,  -fa  of  the  United  States  internal 
revenue  receipts  from  the  tax  on  oleomargarine  amounted 
to  $1,325,021.40.     Find  the  total  receipts  from  this  source. 

15.  The  mark  is  the  unit  of  money  in  Germany  ;    f^  of 
its  value  in  our  currency  is  42  mills.     Express  the  value 
of  a  mark  in  dollars. 

16.  The  yen  is  the  standard  of  value  in  Japan ;  -^  of 
its  value  is  equivalent  to  4  cents  and  2  mills.     Express  in 
dollars  the  value  of  the  yen. 

17.  In   Venezuela,  the  monetary  unit  is  the  Bolivar  ; 
|  of  its  value  is  equivalent  to  $.1158.     Find  its  value  in 
cents. 

18.  Thirty-two  thirty-fifths  of  a  meter  is  very  nearly 
equivalent  to  1  yd.     Express  the  value  of  a  meter  in  yards. 


196  ADVANCED  BOOK  OR  ARITHMETIC 

THE  LANGUAGE  OF  MATHEMATICS,  RATIO,  PROPOR- 
TION,  PARTNERSHIP 

By  mathematics  is  understood  those  branches  of  knowl- 
edge which  deal  with  quantity.  Arithmetic,  algebra, 
geometry,  surveying,  etc.,  are  included  in  the  term  mathe- 
matics. 

Mathematics  has  a  language  of  its  own. 

The  word  eight  conveys  a  definite  idea  to  the  mind ;  the 
sign  or  symbol  8  conveys  the  same  idea.  The  words 
eight  squared  convey  a  definite  idea  to  the  mind  ;  the 
symbol  82  conveys  the  same  idea.  The  words  three  fourths 
of  sixteen  convey  an  idea ;  the  symbols  f  X  16  convey  the 
same  idea.  Similarly,  the  words  the  quotient  of  seventy- 
two  divided  by  eight  convey  an  idea  ;  the  symbol  ^-  con- 
veys the  same  idea. 

Letters  may  represent  numbers.  Thus,  a,  6,  c,  x,  y,  z, 
etc.,  may  each  represent  any  number  whatever.  The 
product  of  the  numbers  represented  by  letters  is  indicated 
by  writing  the  letters  in  succession,  one  after  the  other. 
Thus,  abc implies  the  continued  product  of  a,  6,  and  c.  lip 
stands  for  principle,  r  for  rate,  t  for  time  in  years,  and  i  for 
interest,  the  rule  for  computing  interest  is  given  by  the 
relation  prt  =  { 

In  like  manner  the  rule  for  computing  the  area  of  a  rec- 
tangle may  be  expressed  by  the  relation 

F=la, 

F  stands  for  area,  b  for  base,  and  a  for  altitude.     A  num- 
ber written  before  a  letter  indicates  multiplication. 
Thus,  5  a  means  5  times  a.     5  a  is  then  a  short  way  of 
writing  a  +  a  +  a  +  a  +  a. 

46  is  a  short  way  of  writing  b  -f  b  +  b  +  b. 


THE  LANGUAGE  OF  MATHEMATICS  197 

02  is  a  short  way  of  writing  a  x  a  or  aa. 

03  is  a  short  way  of  writing  a  x  a  x  a  or  aaa. 

04  is  a  short  way  of  writing  0x0x0x0  or  aaaa. 

05  is  a  short  way  of  writing  0x0x0x0x0  or  aaaaa. 
The  number  denoting  how  many  times  a  number  is 

added  is  called  a  coefficient.  The  coefficient  of  the  expres- 
sion 9  b  is  9. 

The  expression  a  +  b  stands  for  the  sum  of  any  two 
numbers.  The  expression  a  —  b  stands  for  the  number 
which  when  added  to  b  gives  0,  or  in  other  words,  the 
remainder  obtained  when  b  is  subtracted  from  a. 

Example  1.  If  a  =  5,  b  =  3,  what  is  the  value  of  0  +  6  ? 
a-b  =  ?  4a  =  ?  36  =  ?  20-36=? 

SOLUTION.    0  +  6  =  5  +  3  =  8.     0-6  =  5-3  =  2. 

40  =  4x5  =  20.  36  =  3x3=9.  20-36  =  2x5-3 
x3  =  l. 

Example  2.  If  a  =  7,  what  is  the  value  of  a2?  03?  3  a2? 
4  a3? 

a2  =  axa==7x7  =  49.     aB==axaxa==j  x 7x7  =343. 

3  02  =  3x0x0=  3x7  x7=147.  403  =  4x0x0x0 
=  4x7x7x7  =  1372. 

EXERCISE   103 

If  0=4,  what  is  the  value  of  40?  70?  110?  130? 
170?  190?  270?  |0?  ^0? 

If  0=5,  what  is  the  value  of  02?  03?  04?  202?  302? 
203?  402?  0  +  02?  02  +  03? 

If  0  =  3,  6  =  2,  what  is  the  value  of  0  +  6?  0  —  6? 
20-f6?  0+26?  20-6?  20-36?  30-26? 

If  0  =  6,  6  =  3,  what  is  the  value  of  60  +  36?  30  +  56? 
60-36?  50-106?  70-56?  302?  02+62?  02-62? 

If  #  =  5,  y  =  6,  what  is  the  value  of  xy?  2xy?  oxyl 
*V?  xy?  Ja? 


198  ADVANCED   BOOK  OF^  ARITHMETIC 

If  x  =  9,  y  =  4,  what  is  the  value  of  2  x*  -  y*  ? 

?    2z2  +3 2/2?    ?/2 -^? 
If  a  =  10,  6  =  7,  find  the  value  of  ab,  5  «6,  a6  +  a2,  2  a6 
+  62,  ab  +  2b*. 

EXERCISE   104 

1.  What  is  the  sum  of  two  times  a  number  and  three 
times  the  same  number?    What  is  the  sum  of  2x  and  3#? 

2.  What  is  the  sum  of  4 x  and  3  x?    of  8  a  and  3  a?    of 
56  and  26?   of  66  and  4  6? 

3.  What  is  the  difference  between  8x  and  3x?    6x  and 
2x?    116  and  76?    Saanda? 

4.  Add  5  x  and  7  # ;  4  #  and  9  # ;    96  and  66;  10  y  and 

6  y ;   12  x  and  4  #. 

5.  Subtract  4  #  from  9  x ;  8  #  from  14  #;   9#froml6#; 

7  x  from  13  x ;  5  a6  from  8  ab. 

6.  Find  the  difference  between  11  y  and  2y;  5a6  and 
a6  ;  7  ab  and  4  a6  ;  12  a6  and  2  #6. 

Every  sentence  conveys  a  thought.  (1)  The  sum  of  3 
and  4  is  7.  This  sentence  is  expressed  in  the  language 
of  mathematics  as  follows :  3  +  4  =  7.  (2)  Write  another 
sentence :  The  difference  between  18  and  7  is  11.  This 
sentence,  written  in  mathematical  language,  is  18  —  7  =  11. 
(3)  Write  a  third  sentence  :  Two  thirds  of  27  is  18.  In 
mathematical  language  this  sentence  is  written  |  x  27=  18. 
The  statements  (1),  (2),  (3),  are  called  equations.  An 
equation  is  a  statement  in  symbols  that  two  expressions 
are  equal  to  each  other. 

The  part  of  an  equation  to  the  left  of  the  sign  of  equal- 
ity is  called  the  first  member  of  the  equation ;  the  part  of 
an  equation  to  the  right  of  the  sign  of  equality  is  called 
the  second  member  of  the  equation. 


THE   LANGUAGE   OF  MATHEMATICS  199 

What  is  the  product  of  a  and  a?     a  x  a  =  a2. 
What  is  the  product  of  a  and  a2?     a  x  a2  =  a  x  a  x  a 
=  a3. 

What  is  the  product  of  a2  and  a3?     #2  =  ax# 
a  x  a  x  a. 

.-.  a2  x  a3  =  (a  x  a)  x  (a  x  a  x  a)  =  a5. 
What  is  the  product  of  a4  and  a3  ? 
a4=axaxaxa. 
a3  =  a  x  a  x  a. 

.'.  a4  x  a3  =  (a  X  a  x  a  x  a)  x  (a  x  a  x  a)  =  a7. 
What  is  the  product  of  4  a2  and 
4  a2  =  4  x  a  x  a. 


.'.4a2x5a3=:4xaxax5x#x  «  x  a 
=4x5xaxa  x  a  x  a  x  a 
=  20  a5.  (Associative  Law.) 

EXERCISE   105 

1.  a  x  2  a  =  ?  7.  3  a  x  5  a3  =  ?  13.  4  62  x  3  b  =  ? 

2.  2axa2=?  8.  9a2x2a3=?  14.  553x453  =  ? 

3.  3  a2  x  a  =  ?  9.  4  a3  x  a2  =  ?  15.  2  b2  x  5  6  =  ? 

4.  3a2x2a2=?  10.  5^x4z  =  ?  16.  4*/2x3#3=? 

5.  4  a  x  a3  =  ?  11.  6  x2  x  :r3  =  ?  17.  7  z2  x  5  a  =  ? 

6.  7  a2  x  2  a2  =?  12.  5  5  x  62  =  ?  18.  4  y  x  8  53  =  ? 

What  is  the  quotient  when  a3  is  divided  by  a? 

o  a3      a  x  a  x  a  o 

ad  -^  a  =  —  =  --  =  a  x  a  =  a*. 
a  a 

Here  cancellation  is  utilized. 
What  is  the  quotient  of  8  a3  by  2  a? 
4 

o         8  a3      $  x  a  x  a  x  a      4 
2a=  ---  =  r—  -  =  4xaxa 

2a  $  xa 


200  ADVANCED  BOOK  OF  ARITHMETIC 

EXERCISE   106 

Find  the  following  quotients: 

i     *±          6    ?1^     u     22o_6  16     §5^             60^ 

2  a'                 7  a2'              2  a2'  Sz3'             10  fc5' 

9a                28a*              16  a»  39^      gg     26^ 

3  '            '    14 a2'              4a3'  13 a;'              13 a;' 
12  a              18  a*              32a«  42 «»              33ft2 

3.     — 8.      — — -.       13.                  .  18.              s  •       2d.                 . 

4  o  a6                Ib  a*  ba^*                116 
16  a2             24  a4              24  a5  45  z4              48  J4 


400                         Q3                         O  1^J7» 

tt                 o  a                  o  d                  \j  x  J-O  o 

12  a3             11  a4              25  «4              50  x5  46  55 

5. 10.     .  15.      -=— x-.  20.     -.  25. 


23  62' 


RATIO 


The  ratio  of  one  number  a  to  another  number  b  is  the 
quotient  obtained  by  dividing  a  by  b. 

The  ratio  of  a  to  b  is  written  a  :  b.     When  the  quotient 

a  -r-  b  is  written  -,  the  expression  -  is  a  fraction. 
b  b 

The  ratio  b  :  a  is  called  the  inverse  ratio  of  a  to  b. 

EXERCISE   107 

1.  What  is  the  ratio  of  2  ft.  to  6  ft.? 

2.  What  is  the  ratio  of  4  in.  to  1  yd.?  of  3  in.  to  1  yd.? 
of  1  yd.  to  1  rd.?  of  l  rd.  to  1  rd.? 

3.  What  is  the  ratio  of  80  A.  to  1  sq.  mi.?  of  120  A. 
to  1  sq.  mi.?  of  l  A.  to  2  A.? 

4.  What  is  the  ratio  of  the  distance  traveled  by  two 
trains  in  the  same  time,  if  the  rate  of  the  first  train  is  20 
mi.  per  hour,  and  the  rate  of  the  second  train  is  30  mi. 
per  hour  ? 


THE  LANGUAGE    OF  MATHEMATICS  201 

5.  If  A  walks  at  the  rate  of  2J  mi.  per  hour,  and  B 
walks  at  the  rate  of  5  mi.  per  hour,  what  is  the  ratio  of 
A's  time  to  B's  time  in  going  any  given  distance  ? 

6.  What  is  the  ratio  of  the  time  that  8  men  take  to  do 
a  piece  of  work  to  the  time  that  6  men  take  to  do  the  same 
piece  of  work  ? 

7.  If  you  ride  in  a  carriage  at  the  rate  of  7  mi.  an  hour 
and  walk  back  the  same  distance  at  the  rate  of  3  mi.  an 
hour,  what  is  the  ratio  of  the  time  in  the  carriage  to  the 
time  walking  ? 

8.  What  is  the  ratio  of  the  price  of  7  Ib.  of  sugar  to  the 
price  of  10  Ib.  of  sugar  of  the  same  kind  ? 

9.  What  is  the  ratio  of  the  work  done  by  6  men  to  the 
work  done  by  9  men  ? 

10.  What  is  the  ratio  of  the  time  that  9  men  take  to  do 
a  piece  of  work  to  the  time  that  6  men  take  to  do  the 
same  work  ? 

11.  I  can  buy  two  kinds  of  matting  for  40^  and  50^  a 
yard  respectively.     If  I  spend  the  same  amount  of  money 
in  the  purchase  of  the  two  kinds  of  matting,  what  is  the 
ratio  of  the  number  of  yards  of  matting  of  the  first  kind 
to  the  number  of  yards  of  the  second  kind  bought  ? 

12.  Divide  15  in  the  ratio  2  :  3. 

13.  Divide  20  in  the  ratio  3 :  7. 

14.  Divide  f  1  in  the  ratio  18  :  7. 

15.  Divide  1  mi.  in  the  ratio  7  :  9. 

16.  Divide  1  in  the  ratio  9  : 11. 

17.  Divide  1  gal.  in  the  ratio  1 :  3. 

18.  Divide  $'1  in  the  inverse  ratio  9  : 16. 

19.  Divide  22  yd.  in  the  inverse  ratio  3  :  8. 

20.  Divide  $1000  in  the  inverse  ratio  3 :  5. 


202  ADVANCED   BOOK  OF  ARITHMETIC 

PROPORTION 

A  statement  indicating  that  two  ratios  are  equal  is 
called  a  proportion. 

Illustrations,      2:3  =  4:6.       (1) 
9:15  =  12:20.       (2) 

Statement  (2)  is  a  proportion  because  the  value  of  the 
first  ratio  is  f ,  and  the  value  of  the  second  ratio,  i.e.  12  :  20, 
is  also  ^. 

Statement  (2)  may  read  9  is  as  large  compared  with 
15  as  12  is  compared  with  20. 

The  first  and  fourth  terms  of  a  proportion  are  called 
the  extremes,  and  the  second  and  third  terms  are  called 
the  means,  of  the  proportion. 

In  a  proportion  the  product  of  the  extremes  is  equal  to  the 
product  of  the  means. 

Let  a :  b  =  c :  d  be  any  proportion  whatever.  Then 
ad  =  be. 

PROOF.     7  =  -;-     Multiply  each  member  by  Id  and  get 
b      d 

abd      cbd  *,  •,-,   ,.          7      z 

— ;—  = .  *.  by  cancellation  ad  =  be. 

b         d 

This  property  of  a  proportion  enables  us  to  find  any 
term  of  a  proportion,  if  three  of  the  terms  of  the  propor- 
tion are  known. 

The  proportion  a:b  =  c:d  is  sometimes  written  a:  b 
:  :c:  d.  The  double  colon  used  as  a  sign  of  equality  is 
now  rapidly  becoming  obsolete. 

Example  1.     Find  x  in  the  proportion  x  :  4  =  9  :  6. 

SOLUTION.  The  product  of  the  extremes  is  equal  to 
the  product  of  the  means. 

.•.63=36.  3=6. 


PROPORTION  203 

Example  2.     Find  x  in  the  proportion  10  :  35  =  x  :  42. 
SOLUTION.     Since  the  product  of  the  means  is  equal  to 
the  product  of  the  extremes, 

35  x  =  10  x42. 

Two  numbers  which  vary  directly  are  said  to  be  directly 
proportional.  Two  numbers  which  vary  inversely  are  said 
to  be  inversely  proportional. 

EXERCISE   108 

If  x  stands  for  the  unknown  term  in  each  of  the  follow- 
ing proportions,  find  it : 

1.  2:3=6:  x.  13.  57  : 133  =  x  :  126. 

2.  3:  4=  6:  a?.  14.  68  :  85  =  x  :  75. 

3.  15:25  =  12:z.  15.  36  :  x  =  52  :  65. 

4.  12:20  =  18:z.  16.  28:^=36:63. 

5.  14:21  =  ^:27.  17.  27:  a  =  15:  50. 

6.  21:  27  =  a;:  45.  18.  15:^=21:77. 

7.  35:  84  =  z:  72.  19.  28:  a  =36:  81. 

8.  20:48  =  z:96.  20.  25:  x  =  45:  72. 

9.  16:24  =  ^:33.  21.  35:^=30:48. 

10.  20:  32  =  x:  72.  22.    x:  81  =  16:  72. 

11.  25:45  =  ^:99.  23.    x:  99  =  26: 117. 

12.  45  : 126  =  x :  154.  24.    x  :  65  =  24  :  52. 

25.    x  :  112  =  45  :  144. 

Example  1.  If  7  bu.  of  wheat  cost  $5.25,  find  the  cost 
of  11  bu.  of  wheat  at  the  same  rate. 

SOLUTION.  It  is  reasonable  to  assume  that  the  price  of 
11  bu.  of  wheat  is  greater  than  the  price  of  7  bu.  of  wheat. 
.-.  the  price  of  11  bu.  of  wheat  =  -U  of  $5.25=  $8.25. 


204  ADVANCED  BOOK  OF^  ARITHMETIC 

Example  2.  If  12  men  pave  a  street  in  15  da.,  how 
long  will  it  take  9  men  to  pave  a  street  of  the  same  area  ? 

SOLUTION.  It  will  take  9  men  longer  than  it  takes  12 
men.  /.  the  time  9  men  take  =  -^  of  15  da.  =  20  days. 

To  solve  a  problem  in  proportion,  find  first  the  relation 
of  the  answer  to  the  quantity  of  the  same  kind  as  the 
answer  given  in  the  problem.  Second,  multiply  this 
quantity  by  a  fraction,  proper  or  improper,  according  as 
the  answer  is  less  or  greater  than  it. 

EXERCISE   109 

1.  If  20  men  earn  $450  in  a  given  time,  how  much  will 
30  men  earn  in  the  same  time  ? 

2.  If  15  bu.  of  corn  cost  $7.20,  what  will  48  bu.  of 
corn  cost? 

3.  If  12  A.  of  land  cost  $456.90,  what  will  16  A.  of 
the  same  land  cost  ? 

4.  If  4  men  can  do  a  piece  of  work  in  15  da.,  how  long 
will  it  take  6  men  to  do  an  equal  amount  of  work  ? 

5.  If  18  head  of  cattle  cost  $1450,  what  will  27  head 
of  cattle  cost  at  the  same  rate  ? 

6.  If   a  train  goes  400  mi.  in  12  hr.,  how  long  will 
it  take  to  go  560  mi.  ? 

7.  If  8  masons  build  a  wall  in  15  da.,  how  long  will 
it  take  6  masons  to  build  a  wall  of  the  same  size  ? 

8.  If    18    horses  consume  14  bu.  of    corn  in  a  week, 
how  much  will  24  horses  consume  in  the  same  time  ? 

9.  If  18  horses  plow  a  tract  of  land  in  13  da.,  how 
long  will  it  take  26  horses  to  plow  the  same  tract  ? 

10.  How  long  will  it  take  126  sheep  to  eat  a  quantity 
of  feed  which  will  last  105  sheep  30  da.  ? 


COMPOUND  PROPORTION  205 

11.  A  garrison  consisting  of  1200  men  has  provisions 
for  16  da.     How  many  men  must  be  sent  away  so  that 
the  provisions  may  last  24  da.  ? 

12.  A  garrison  consisting  of  1400  men  has  provisions 
for  27  da.     If   the  garrison  is  reenforced    by  400  men, 
in  how  many  days  will  the  provisions  be  consumed? 

13.  If    I  can  buy  a   dozen  turkeys  for   $20.50,   how 
many  turkeys  can  I  buy  for  $30.75? 

14.  If  the  interest  on  $750  for  4  mo.  is  $12.50,  what 
is  the  interest  on  $39.60  for  the  same  time? 

15.  If  an  arc  of  12"  on  the  40th  parallel  of  latitude  is 
933.92  ft.,  find  the  length  of  1°  on  the  40th  parallel  of 
latitude. 

16.  If  an  arc  of  30'  on  the  circumference  of  a  wheel  is 
1^  in.,  find  the  length  of  the  circumference  of  the  wheel. 

17.  A  fly  wheel  63  ft.  in  circumference  makes  150  revo- 
lutions per  minute.    Find  the  velocity  of  its  rim  per  second. 

18.  A  train  is  running  at  50  miles  an  hour.     This  speed 
is  25%  greater  than  usual.     Find  its  usual  speed. 

COMPOUND  PROPORTION 

If  the  product  of  the  corresponding  terms  of  two  or 
more  ratios  are  taken,  the  ratio  of  the  resulting  products 
is  called  the  ratio  compounded  of  these  ratios.  For 
example,  the  ratio  compounded  of  the  ratios  2  :  3,  4  :  5, 
7:8,  is  the  ratio  2x4x7:3x5x8,  or  56  :  120,  or 
7  :  15. 

A  proportion  in  which  the  final  result  depends  upon 
a  ratio  compounded  of  two  or  more  ratios  is  called  a 
compound  proportion. 

A  concrete  example  may  give  a  clearer  conception  of 
compound  proportion  than  any  formal  definition. 


206  ADVANCED  BOOK  OF  ARITHMETIC 

Example  l.  If  15  men  mow  90  A.  in  12  da.,  how 
many  acres  will  12  men  mow  in  14  da.? 

SOLUTION.  The  12  men  in  a  given  time  will  mow  less 
than  15  men  in  the  same  time.  .*.  the  12  men  in  12  da. 
will  mow  If  of  90  A.  But  the  12  men  in  14  da.  will 
mow  more  than  this  quantity.  /.  12  men  in  14  da.  will 
mow  if  of  if  of  90  A.  =  if  of  90  A.  =  84  A. 

Example  2.  If  24  men  build  a  house  in  18  da.  of  10  hr. 
each,  how  many  men  will  it  take  to  build  the  same  house 
in  30  da.  of  8  hr.  each? 

SOLUTION.  Step.  1.  It  will  take  fewer  men  to  build 
a  house  in  30  da.  than  it  will  take  to  build  it  in  18  da. 
of  the  same  length. 

.*.  the  number  of  men  it  will  take  to  build  the  house 
in  30  da.  of  10  hr.  each  =  if  of  24  men. 

Step  2.  More  men  are  needed  when  they  work  8  hr.  a 
day  than  when  they  work  10  hr.  a  day. 

.*.  the  number  of  men,  in  30  da.  of  8  hr.  each,  required 
to  build  the  house  =  -^  of  ^|  of  24  men=  18  men. 

EXERCISE   110 

1.  If  12  horses  plow  84  A.  in  6  da.,  how  many  acres 
will  16  horses  plow  in  4^  da.? 

2.  If  14  men  pave  a  street  200  ft.  long  in  8  da.,  how 
many  feet  will  12  men  pave  in  7  da.  ? 

3.  If  a  man  earns  $117  in  3  mo.  working  6  hr.  a  day, 
how  much  will  he  earn  in  5  mo.  working  8  hr.  a  day  ? 

4.  A  garrison  of  3650  men  consumed  in  30  da.  82.3  T. 
of  food.     How  much  food  would  be  required  for  7500 
men  for  1  yr.  at  the  same  rate  ? 


COMPOUND   PROPORTION  207 

5.  If  8  masons  build  in  2  da.  a  wall  40  ft.  long  and 
6  ft.  high,  what  height  of  wall  30  ft.  long  can  they  build 
in  5  da.  ? 

6.  If  21  men  complete  a  piece  of  work  in  8  da.  of  7|- 
hr.  each,  in  how  many  days  of  10  hr.  each  can  18  men 
do  the  same  work  ? 

7.  A  wall  is  to  be  built  in  10  da.  by  30  men.     After 
2  da.   10  men   are   dismissed.      In  what   time  will   the 
remaining  20  men  finish  the  work  ? 

8.  If  4  men  or  6  boys  dig  a  trench  in  12  da.,  in  what 
time  can  2  men  and  9  boys  dig  it  ? 

9.  If  12  men  mow  30  A.  in  3  da.  of  8  hr.  each,  how 
many  hours  a  day  must  16  men  work  to  mow  48  A. 
in  4  da.  ? 

10.  If  the  interest  on  $100  for  1  yr.  is  16,  find  the 
interest  on  1840  for  2  yr.  3  mo. 

11.  If  12  men  working  7  hr.  a  day  earn  1227.50  in  20 
da.,  how  much  will  15  men  earn  in  20  da.,  working  9  hr. 
each? 

12.  If  6  men  mow  f  of  a  meadow  in  4J  da.,  how  long 
will  it  take  8  men  to  mow  the  remainder  ? 

13.  In  10  da.  of  8  hr.  each  9  horses  can  plow  f  of  a 
field.     In  how  many  days  of  9  hr.  each  can  the  remainder 
of  the  field  be  plowed  by  15  horses  ? 

14.  A  marble  block  3  ft.  by  4  ft.  and  5  ft.  in  length 
weighs  5.1  T.     Find  the  weight  of  a  marble  block  7  ft. 
by  3  ft.  and  10  ft.  long. 

15.  A  mason  can  build  3  yd.  of  a  wall  in  15  hr.     How 
long  will  it  take  9  masons  to  build  24  yd.  of  a  wall  whi  a 
is  one-third  higher  than  the  other  wall  ? 


208  ADVANCED  BOOK  OF  ARITHMETIC 


PARTNERSHIP 

NOTE.  Partnerships  are  rapidly  becoming  a  thing  of  the  past.  Those 
partnerships  that  still  survive  are  conducted  on  somewhat  different 
principles  from  the  partnerships  that  existed  prior  to  the  introduction  of 
the  telegraph,  telephone,  and  modern  means  of  rapid  transit. 

Example.  A,  B,  and  C  enter  into  partnership.  A  puts 
in  $840,  B  puts  in  $350,  and  C  puts  in  $2000.  A  with- 
draws from  the  concern  in  5  mo.,  C  in  7  mo.,  and  at  the 
end  of  8  mo.  the  profits  are  divided.  If  the  entire  profit 
is  $450,  how  shall  this  be  divided  among  A,  B,  and  C? 

SOLUTION.  A  has  $840  in  the  concern  for  5  mo.  This 
is  equivalent  to  $4200  for  1  mo. 

B  has  $350  in  the  concern  for  8  mo.  This  is  equiva- 
lent to  $2800  for  1  mo. 

C  has  in  the  concern  $2000  for  7  mo.  This  is  equiva- 
lent to  $14,000  for  1  mo. 

The  profits  will  be  divided  in  proportion  to  the  num- 
bers 4200,  2800,  14,000,  or  in  proportion  to  the  numbers 
3,  2,  10,  since  1400  divides  each  of  them. 

3  +  2  +  10  =  15. 

.•.  A's  share  =  -^  of  the  profits  =  -f%  of  $450  =  < 
B's  share  =  T2^  of  the  profits  =  T2^  of  $450  =  I 
C's  share  =  ^  of  the  profits  =  jf  of  $450  =  $300. 


EXERCISE  111 

l.  A,  B,  and  C  enter  into  partnership  with  capitals  of 
$3000,  $3750,  and  $4500  respectively.  At  the  end  of 
the  year  they  divide  among  themselves  a  profit  of  $3000. 
Find  each  person's  share. 


PARTNERSHIP  209 

2.  Two    partners,  A  and  B,  invest   $600  and  $1125. 
A's  money  remains  in  the  business  6  mo.,  and  B's  8  mo. 
If  they  make  a  profit  of  $2100,  find  each  person's  share. 

3.  Two  men  rent  a  pasture  for  $171  ;  one  puts  in  the 
pasture  30  cattle  for  30  da.,  and  the  other  45  cattle  for 
18  da.     How  much  rent  should  each  pay  ? 

4.  A   and  B    enter   into   partnership.     A's  capital    is 
$200  more   than    B's.     Out  of   a  profit  of  $640,  B  gets 
$280.     Find  A's  and  B's  capital. 

5.  A  and  B  enter  into  a  partnership,  A  contributing 
$6400  and  B  $7200.     At  the  end  of  3  mo.  A  withdraws 
$1600,  and  at  the  end  of  5  mo.   B  withdraws  $1440.     C 
then  enters  into  the  partnership  with  a  capital  of  $4800. 
Seven   months  later  a  gain    of   $2154  is  divided  among 
them.     Find  each  person's  share. 

6.  A,   B,  and  C  enter  into  partnership.     A   puts   in 
$1000,  B  $1200,  and  C  $1800.     At  the  end  of  3  mo.  C 
withdraws,  and  at  the  end  of  10  mo.  B  withdraws.     At 
the  end  of  a  year  the  profits  are  divided.     If  C  gets  $135, 
how  much  do  A  and  B  receive  ? 

7.  Two  men  form   a  partnership.     Their  capitals  are 
in  the  ratio  2 : 3.     After  6  mo.  the  first  man  increases  his 
capital  by  J  of  itself,  and  the  second  man  diminishes  his 
capital  by  J  of  itself.     After  6  mo.  more  they  divide  their 
profits,  amounting  to  $1450.     Find  each  partner's  share. 

8.  A  cistern  66'  x  27'  6"  x  10' will  hold  enough  water  to 
irrigate  2J  A.  of  land  to  the  depth  of  2  inches.    How  many 
acres  will  a  cistern  77'  x  41'  3"  x  12'  6"  irrigate  to  the 
depth  of  3J  inches  ? 

9.  If  A  pays  |-  of  the  cost  of  irrigation  when  the  rate 
charged  is  $4  an  acre  to  the  depth  of  one  inch,  find  his 
share  of  the  cost. 


210       ADVANCED  BOOK  OF  ARITHMETIC 

PERCENTAGE 

A  per  cent  of  a  number  implies  a  fraction  of  the  num- 
ber having  100  for  denominator.  Thus,  5  per  cent,  5%, 
yj-g-,  and  .05  are  four  ways  of  expressing  the  same  fact. 

The  per  cent  equivalents  of  the  following  fractions 
should  be  thoroughly  fixed  in  mind : 

i'  i*  f>  i>  I'  i'  t>  t'  i>  <b  f>  i>  t>  f'  s>  iV- 
Example  l.    The  total  value  of  imports  into  this  coun- 
try through  the  Atlantic  ports  for  the  year  1906    was 
$974,562,800;  of  this  75.35%   came  through  New  York 
City.     Find  the,  value  of  the  imports  through  this  city. 

SOLUTION.     $974,562,800  x  7-^~  =  $9,745,628  x  75.35 

=  $734,333,069.80,  value  of  imports  through  New  York. 
As  75.35%  is  correct  to  four  figures  only,  the  result  is 
not  likely  to  be  correct  to  more  than  four  figures.  To 
get  four  figures  multiply  97.456  millions  by  7.535  by  the 
contracted  process  explained  on  page  191. 

EXERCISE   112 

1.  Write    the   equivalent   per   cents  of  the  following 
decimals : 

.04,  .08,  .075,  .0525,  .1666f. 

2.  Express  as  decimals  the  following  per  cents : 

41%,  15%,  121%,  621%,  6i%,  3|%. 

3.  Find  5  %  of  each  of  the  following  numbers  : 

2151,  366.7,  689.5,  7.188,  12.469. 

4.  Find  6  %  of  each  of  the  following  numbers  : 

5262,  520.7,  2.66,  3.097,  6.41,  .783. 

5.  Find  4|  %  of  each  of  the  following  numbers : 

4150,  1418,  7120,  43.43,  53.17,  2.42. 


PERCENTAGE  211 

6.  The  Engineer's  Year  Book  for  the  year  1906  gives 
the  cost  of  railway  construction  in  England  as  $194,660 
per  mile.     The  per  cents  of  cost  were  as  follows : 

Land  10  Permanent  way  11| 
Fencing  1J  Sidings  3 
Earthworks  24  Junctions  1 
Tunnels  12  Stations  6-| 
Viaducts  and  bridges  17  Maintenance  J 
Accommodation  works  2  Legal  and  engineer- 
Culverts  5  ing  expenses  6 

Find  the  cost  of  each  of  the  above  items  of  expense. 

7.  The  value  of  the  total  imports  to  the  United  States 
for  the   year   1906  was   $1,226,560,000.     Of  this  value 
79.45%  came  through  the  Atlantic  ports,  4.42%  through 
the  Gulf  ports,  1.38%  through  the  Mexican  border  ports, 
5.41%    through   the    Pacific   ports,   7.97%    through   the 
northern  border  ports,  1.37%  through  the  interior  ports. 

Find  the  value  of  the  imports  through  each  of  these 
divisions. 

8.  The  value  of  the  total  exports  of  the  United  States 
for  the  year  ending  June  30,  1906,  was  $1,743,860,000. 
The  per  cents  of  total  value  by  principal  customs  districts 
were  as  follows : 

New  York          34.81  Savannah  3.72 

Boston  5.66  Puget  Sound  2.82 

New  Orleans        8.63  Detroit  2.02 

Galveston  9.54  Buffalo  Creek  1.72 

Mobile  1.25 

Philadelphia        4.73  Newport  News  1.15 

Baltimore  6.31  Wilmington  1.06 

San  Francisco      2.29  Pensacola  1.06 

Find  the  values  of  the  exports  through  these  cities. 


212 


ADVANCED  BOOK  OF  ARITHMETIC 


Given  a  quantity,  to  find  its  value  when  decreased  by  a 
per  cent  of  itself. 

Example  1.  In  the  year  1906  the  state  of  Ohio  produced 
11,562,500  Ib.  of  wool ;  this  shrunk  50  %  from  scouring. 
Find  the  number  of  pounds  of  scoured  wool. 

SOLUTION.     100  %  -  50  %  =  50  %  =  J. 
11,562,500x1=5,781,250. 

Am.  5,781,250  Ib. 

EXERCISE   113 

l.  The  wool  production  and  per  cent  of  shrinkage  from 
scouring  for  the  year  1906,  as  given  by  the  Bulletin  of 
National  Association  of  Wool  Manufacturers,  for  the 
states  named  are  as  follows  : 


STATE 

NUMBER  POUNDS 
UNWASHED 

PER  CENT  OF 
SHRINKAGE 

Michigan   

9,450,000 

50 

2,450,000 

52 

568,750 

40 

35,815,000 

65 

Wyoming  

32,849,750 

68 

Idaho     

16,905,000 

67 

Oregon  

15,300,000 

70 

13,125,000 

67 

Utah      

12,350,000 

65 

New  Mexico  

15,950,000 

62 

Colorado     

9,450,000 

67 

Arizona      

4,420,000 

66 

Texas     

9,360,000 

66 

4,887,500 

70 

Find  the  number  of  pounds  of  scoured  wool  produced 
in  each  of  the  states. 


PERCENTAGE  213 

Given  a  per  cent  of  a  number,  to  find  the  number. 
Example  l.    During  the  month  of  January  the  average 
daily  attendance  of  a  school  was  414.     This  number  was 
92  %  of  the  school  enrollment.     Find  the  number  enrolled. 
SOLUTION.     92  %  of  enrollment  is  given. 

100  %  of  enrollment  is  sought. 
.-,  enrollment  =  414  x  -^  =  450,  or 
if  x  stands  for  enrollment, 

414 

.9  a  =  414,  therefore  x  =  -  ^  =  450. 

.  y 

Example  2.    A  dealer  sells  an  article  for  $ 522  at  a  gain 
of  16%.     Find  the  cost  price. 

SOLUTION.     116  %  of  cost  price  is  given. 
100  %  of  cost  price  is  sought. 

/.  cost  price  =  $522  x  —  =  1450,  or 

1.16s  =  $522. 

.'.x  =  $522  -s-1.16  =  $450. 

EXERCISE   114 

1.  Find  the  number  of  which  79  is  4  % . 

2.  In  a  certain  town  60  %  of  the  grown  people  are  mar- 
ried.    If  there  are  2394  married  people,  how  many  grown 
people  are  in  the  town  ? 

3.  A  man  spends  $320  for  board.     This  sum  is  40%  of 
his  income.     Find  his  income. 

4.  A  man  spends  83%  of  his  salary  and  saves  1170. 
What  is  his  salary  ? 

5.  A  lot  is  sold  for  $3380  at  a  gain  of  12f  %.     Find 
the  cost  of  the  lot. 

6.  After  a  discount  of  16f  %  is  given,  a  man  pays  $84 
for  a  bill  of  goods.     Find  the  amount  of  the  bill. 


214 


ADVANCED  BOOK  OF  ARITHMETIC 


7.  The  total  levies  of  ad  valorum  taxes  and  tax  rate 
per  cent  of  assessed  valuation  are  as  follows  in  the  states 
named : 


STATE 

LEVY 

RATE  PER  CENT 

Maine       .     

$  6,855,776 

1.95 

Pennsylvania    

58,269,455 

1.49 

South  Carolina           ....          ... 

3  736  344 

1  91 

Kansas     

14  847  136 

4.09 

Tennessee     .     .     

7,626  068 

1.88 

9,002,727 

3.45 

Texas       

13,683,526 

1.34 

Find  the  assessed  valuation  of  property  in  each  of  these 
states. 

To  express  one  number  as  a  percentage  of  another  number. 

Example  1.  The  foreign  population  of  Danish  extrac- 
tion according  to  the  United  Census  of  1890  and  1900  was 
132,543  and  153,805. 

Find  the  increase  per  cent  during  the  ten  years. 

SOLUTION.    153805  -  132543  =  21262,  increase. 
21262 


132543 

21262 
132543 


=  fraction  the  increase  is  of 
population  in  1890. 

£,=16.04%.         In- 


16.04 


crease  per  cent. 


132^)2126200 

132543 

80077 

79526 

551 

530 


As  the  divisor  contains  6  figures  and 
the  quotient  is  required  to  4  figures, 
for  each  quotient  figure  cut  off  one 
from  the  divisor  instead  of  annexing  a 
cipher. 


PERCENTAGE 


215 


Example  2.  A  dealer  buys  goods  at  a  discount  of  40  % 
off  the  list  price,  and  sells  them  at  16  %  off  the  list  price. 
Find  his  gain  per  cent. 

SOLUTION 

Cost  price  to  dealer  =  60  %  of  list  price  (100  %  -  40  %). 
Selling  price  of  dealer  =  84  %  of  list  price  (100  %  -  16  %). 

Gain  =  24  %  of  list  price. 
|-^  x  100  %  =  rate  per  cent  of  gain.     Ans.  40  °/0. 

EXERCISE   115 

1.  The  railway  mileage  of  the  world  January  1,  1906, 
as  given  by  a  German  statistician  was  as  follows : 


COUNTRY 

MILES 

COUNTRY 

MILES 

Europe  

192,251 

North  America 

253,098 

Asia        

50,593 

South  America       .     . 

32,859 

Africa    

16,538 

Australasia   .... 

17,441 

Find  the  per  cent  of  the  total  railway  mileage  in  each 
of  the  six  continents. 

2.  The  foreign-born  population  of  the  United  States 
by  countries  for  the  years  1890  and  1900  was  as  follows  : 


COUNTRY 

1890 

1900 

COUNTRY 

1890 

1900 

Austria    .     . 
England  .     . 
France      .     . 

123,270 
909,090 
113,174 

275,910 
840,513 
104,197 

Germany 
Ireland    .     . 
Scotland 

2,785,000 
1,871,500 
242,200 

2,663,000 
1,615,500 
233,500 

Find  the  rate  per  cent  of  increase  or  decrease. 

3.  A  dealer  buys  goods  at  a  discount  of  40  %  off  the 
list  price,  and  sells  them  at  2  %  below  the  list  price.  What 
per  cent  of  profit  does  he  make? 


216  ADVANCED  BOOK  OF  ARITHMETIC 

4.  Eggs  are  bought  at  the  rate  of  5  for  4^,  and  sold 
at  the  rate  of  4  for  5^.     What  per  cent  of  profit  is  made? 

5.  A  lot  is  sold  for  $1560  at  a  profit  of  $120.     Find 
the  rate  per  cent  of  profit. 

6.  Meat  is  sold  at  18^  per  pound  at  a  profit  of  20%. 
Find  the  cost  price  per  pound. 

7.  If  the  butcher  has  to  pay  1^  per  pound  more  for  the 
meat,  how  must  he  sell  it  to  make  a  profit  of  25  %? 

8.  A  piano  is  sold  for  $470  at  a  loss  of  6%.     What 
would  the  gain  per  cent  have  been  if  the  piano  had  been 
sold  for  $520? 

9.  A  tradesman  buys  at  a  discount  of  10%,  and  sells 
at  an  advance  of  15  %  on  the  nominal  cost  price.     Find  his 
rate  per  cent  of  profit. 

10.  A  book  costs  the  publisher  60^  for  printing  and 
publishing.     At  what   price  should   he  sell  the  book  in 
order  that  he  may  make  a  profit  of  20%,  after  paying  the 
author  10%  on  the  selling  price? 

11.  What  should  be  the  selling  price  of  an  article  which 
costs  $15,  so  that  a  profit  of  20  %  may  be  made  after  giv- 
ing the  dealer  a  discount  of  10  %  ? 

12.  A  tradesman  marks  his  goods  at  25  %  above  cost, 
but  allows  the  customer  6  %  discount.     What  per  cent  of 
profit  does  he  make  ? 

13.  Tea  is  sold  at  60^  per  pound  at  a  profit  of  33J%. 
If  the  total  gain  is  $15,  how  much  tea  is  sold  ? 

14.  A  man  buys  a  house  for  $4000  which  he  rents  for 
$40   per   month;  his   taxes   are    3  %    on  a   valuation  of 
$  3000.     What  per  cent  does  his  money  yield  ? 

15.  A   merchant   marks   his   goods   20  %    above   cost. 
What  discount  does  he  give  if  he  sells  at  cost  ? 


INTEREST  217 

INTEREST 

Example  1.    Find   the   interest   on  $  670   at  5%   from 
Jan.  14  to  Aug.  10.    . 

MO.     DA. 

SOLUTION.  $670  Aug.  10  =  8    10 

.05  Jan.  14  =  1     14 

6  mo.  =  £  of  1  yr.  2)133  50  =  int.  for  1  yr.  6     26 

16.75  =  int.  for  6  mo. 

20  da.  =  1  of  6  mo.        1.861  =  int.  for  20  da. 

5  da.  =  I  of  20  da.          .465  =  int.  for  5  da. 

1  da.  =  |  of  5  da.  .093  =  int.  for  1  da. 

$19.17  =  int.  for  6  mo.  6  da. 

EXERCISE   116 

Find  the  interest  and  amount  of  : 

1.  1728  for  1  yr.  6  mo.  at  5%. 

2.  $670  for  1  yr.  6  mo.  at  7  %. 

3.  $1260  for  1  yr.  3  mo.  at  8%. 

4.  $385  for  1  yr.  4  mo.  12  da.  at  1%. 

5.  $2750  for  1  yr.  8  mo.  at  3  %. 

6.  $3345  for  1  yr.  4  mo.  at  6%. 

7.  $783  for  1  yr.  1  mo.  10  da.  at  4  %. 

8.  $597  for  1  yr.  4  mo.  24  da.  at  5  %. 

9.  $3000  for  3  mo.  6  da.  at  7%. 

10.  $940  for  1  yr.  4  mo.  at  3%. 

11.  $1800  for  1  mo.  15  da.  at  4%. 

12.  $  2100  for  2  yr.  9  mo.  at  4%. 

13.  $960  for  8  mo.  17  da.  at  1%. 

14.  $2911.25  for  1  yr.  7  mo.  16  da.  at  4%. 

15.  $1857  for  1  yr.  5  mo.  18  da.  at  5%. 

16.  $2775  from  May  1  to  Dec.  19  at  4  %. 


218  ADVANCED  BOOK  OF  ARITHMETIC 

17.  $1770  from  Jan.  10  to  Oct.  5  at  4J  %. 

18.  $1975.14  from  Feb.  8  to  Nov.  1  at 

19.  $1218  from  March  6  to  Nov.  1  at 

20.  $1788  from  Feb.  14  to  Dec.  20  at  7-|  %. 

EXACT  INTEREST 

Interest  reckoned  on  the  basis  of  365  days  to  the  year 
is  called  exact  interest.  Exact  interest  is  used  by  the 
United  States  government  and  sometimes  in  business 
transactions. 

Example.  Find  the  exact  interest  on  $2384.50  from 
Jan.  12  to  July  5  at  5%. 

SOLUTION.  From  Jan.  12  to  July  5  there  are  (19  -f  28 
+  31  +  30  +  31  +  30  +  5).  days  =  174  days. 

$2384.50  x  .05  x'i£f  =  exact  interest. 

$2384.50  x. 05x174 

-— —          -  =  $56.84,  nearly. 

obo 

EXERCISE  117 

Find  the  exact  interest  on  : 

1.  $913  from  Jan.  4  to  Feb.  4  at  5%. 

2.  $731.11  from  Jan.  14  to  Jan.  28  at  7%. 

3.  $52.50  from  Jan.  1  to  April  28  at  7  %. 

4.  $2745  from  Feb.  1  to  April  6  at  5  %. 

5.  $1095.80  from  March  6  to  June  7  at  5%. 

6.  $1911.17  from  March  1  to  May  11  at  7  %. 

7.  $1464.98  from  Jan.  4  to  May  30  at  6  %. 

8.  $10565.65  from  May  13  to  June  25  at  4  % 

9.  $834  from  Feb.  5  to  July  12  at  11  %. 
10.  $3561.50  for  81  da.  at  5%. 


INVERSE  QUESTIONS  IN  INTEREST  219 

INVERSE   QUESTIONS   IN  INTEREST 

Example  l.    What  principal  will  produce  $78.75  interest 
in  75  days  at  1\%1 

Let  x  denote  the  principal. 


.•.  $#  will  produce  178.75  x  -y~-  in  1  year. 


=  78.75  x 


Example  2.    In  what  time  will  $840  produce  $57.40  in- 
terest at  5  %  ? 

Int.  on  $840  for  1  yr.  at  5  %  =  $42. 

$57.40      • 


11  of  1  yr.  =  11  of  12  mo.  =  4f  mo. 
I  of  a  mo.  =  |  of  30  da.  =  12  da. 
The  time  is  1  yr.  4  mo.  12  da. 

Example  3.    At  what  rate  percent  will  $720  produce 
$42.50  interest  in  1  yr.  2  mo.  5  da.  ? 
$720  will  produce  $42.50  -*-  (1  +  1  +  ^|^)  in  1  yr. 

$42  50 
$  1  will  produce  -in-  -*-(!+£+  3  to)  in  *  Jr- 


$100  will  produce  100  x'^-(l  +  H  si  o)inlJr-  =  *5- 

.•.  the  rate  is  5%. 

Example  4.  What  principal  will  amount  to  $136.27  in 
1  yr.  3  mo.  15  da.  at  5  %? 

The  interest  on  $1  for  1  yr.  3  mo.  15  da.  is  $.0645|. 

.-.  the  amount  of  $1  for  1  yr.  3  mo.  15  da.  is  $1.0645f. 

.  \  the  number  of  dollars  in  principal  =  $136.27  -*- 
$1.0645f  =  $128,  nearly. 


220  ADVANCED  BOOK  OF  ARITHMETIC 

EXERCISE  118 

What  principal  will  produce: 

1.  $60  in  11  yr.  at  8%? 

2.  $120  in  2yr.  at  5  %  ? 

3.  1135  in  1  yr.  6  mo.  at  9%? 

4.  $36  in  3  yr.  at  5%? 

5.  $144  in  1  yr.  4  mo.  at  4J%? 

6.  $12  in  1  yr.  at  4%? 

7.  $21  inl  yr.  at  3|  %? 

8.  $84  in  3yr.  6  mo.  at  3%? 

9.  $16.90  in  2  yr.  2  mo.  at  4  %  ? 

10.  $  42  in  2  yr.  4  mo.  at  4  %  ? 

11.  $25.50  in  6  mo.  at  5 %? 

12.  $5.40  in  4  mo.  at  5 %? 

13.  $6.75  in  9  mo.  at  3%? 

EXERCISE   119 
In  what  time  will: 

1.  $1088.75  produce  $87.10  interest  at  8  %  ? 

2.  $144  produce  $21.60  interest  at  5  %  ? 

3.  $215  produce  $6.45  interest  at  5  %  ? 

4.  $  1160  produce  $278.40  interest  at  6  %  ? 

5.  $810  produce  $56.70  interest  at  7  %  ? 

6.  $312.50  produce  $43.75  interest  at  8  %  ? 

7.  $2220  produce  $216.45  interest  at  S%? 

8.  $1400  produce  $78.75  interest  at  5%? 

9.  $480  produce  $85.50  interest  at  9|  %  ? 

10.  $3835  produce  $345.15  interest  at  8%? 

11.  $1380  produce  $88.55  interest  at  3|%? 

12.  $5400  produce  $267.75  interest  at  7  %  ? 

13.  $7630  produce  $1335.25  interest  at  6  %  ? 


INVERSE  QUESTIONS  IN  INTEREST  221 

EXERCISE  120 
Find  the  rate  per  cent  when  the  interest  on  — 

1.  $750  for  1  yr.  is  $45. 

2.  1928  for  1  yr.  is  1 64. 96. 

3.  $880  for  1|  yr.  is  $79.20. 

4.  $945  for  6  mo.  is  $37.80. 

5.  $828  for  8  mo.  is  $38. 64. 

6.  $1200  for  1  yr.  3  mo.  is  $90. 

7.  $1800  for  9  mo.  is  $67.50. 

8.  $2400  for  8  mo.  is  $64. 

9.  $2500  for  9  mo.  18  da.  is  $100. 

10.  $3000  for  7  mo.  12  da.  is  $90. 

11.  $3750  for  4  mo.  15  da.  is  $112.50 

12.  $2754  for  2  mo.  20  da.  is  $55.08. 

13.  $4846  for  6  mo.  20  da.  is  $121.15. 

14.  $1440  for  7  mo.  10  da.  is  $52.80. 

Find  the  rate  per  cent  when  — 

15.  $1080  amounts  to  $1123.20  in  8  mo. 

16.  $1200  amounts  to  $1270  in  10  mo. 

17.  $1600  amounts  to  $1640  in  6  mo. 

18.  $2460  amounts  to  $2574.80  in  9  mo.  10  da. 

19.  $92  amounts  to  $102.12  in  2  yr. 

20.  $324  amounts  to  $333.72  in  8  mo. 

EXERCISE  121 

What  principal  will  amount  to  — 

1.  $840  in  1  yr.  at  5%  ?    4.    $903  in  1|  yr.  at  5  %  ? 

2.  $749  in  1  yr.  at  7%  ?    5.    $414  in  3  yr.  at  5%  ? 

3.  $645  in  1  yr.  at  7-|  %  ?  6.    $255.30  in  2  yr.  at 


222  ADVANCED  BOOK  OF  ARITHMETIC 

7.  $12,540.45  in  2  yr.  3  mo.  at  4  %? 

8.  $168.35  in  7  mo.  6  da.  at  (  %  • 

9.  $618.67  in  1  yr.  4  mo.  at  5%> 

10.  $646.80  in  8  mo.  at  4%? 

11.  $776.07  in  1  yr.  1  mo.  at  4|  %? 

12.  $481.50  inlyr.  at  7  %? 

13.  $432.55  in  11  mo.  at  8 %? 

14.  $282.75  in  2  yr.  4  mo.  15  da.  at  7|  %? 

15.  $2090.07  in  1  yr.  1  mo.  15  da.  at  8  %? 

16.  $2067.75  in  1  yr.  5  mo.  at  10|%? 

17.  $268.28  in  1  yr.  7  mo.  at  6%? 

18.  $254.25  in  7  mo.  15  da.  at  9|%? 

19.  $25,346.25  in  1  yr.  3  mo.  at  10%? 

20.  $843.70  in  8  mo.  3  da.  at  7|%? 

EXERCISE   122 
REVIEW 

1.  Find  the  interest  on  $4000  for  13  mo.  2  da.  at  9%. 

2.  Find  the  interest  on  $256.30  for  4  mo.  9  da.  at  1%. 

3.  Find  the  interest  on  $30.85  for  11  mo.  6  da.  at  5%. 

4.  Find  the  interest  on  $653  for  2  mo.  16  da.  at  4%. 

5.  Find  the  interest  on  $2105.60  for  84  da.  at  5%. 

6.  Find  the  amount  of  $805  for  10  mo.  at  8%. 

7.  Find  the  amount  of  $507  for  1  yr.  12  da.  at  8%. 

8.  What  principal  will  produce  $20.83  interest  in  5 
mo.  at  5%? 

9.  What  principal  will  produce  $17.50  interest  in  9 
da.  at  5%? 

10.    Find  the  rate  of  interest  when  $500  produces  $2.92 
interest  in  1  mo. 


REVIEW  223 

11.  Find  the  rate  of  interest  when  $250  produces  17.30 
in  5  mo. 

12.  How  much  must  I  invest  at  5%  interest  to  have  an 
annual  income  of  $1200  from  my  investment  ? 

13.  A  man  buys  a  house  and  lot  and  rents  it  for  $40  a 
month.     Taxes  and  insurance  cost  him  $120  a  year.     If 
his  net  receipts  give  him  a  profit  of  6%  on  his  investment, 
find  the  cost  of  the  house  and  lot. 

14.  For  how  long  a  time  must  $3000  be  loaned  at  5% 
to  produce  $20  interest? 

15.  If  I  borrow  $2400  at  1%  interest  and  pay  in  princi- 
pal and  interest  $2456,  how  long  did  I  keep  the  money? 

16.  For  how  long  a  time  must   a   sum  of   money  be 
loaned  at  simple  interest  at  8%   to  produce  in  interest 
-1  of  itself  ? 

17.  For  how  long  a  time  must  a   sum  of   money  be 
loaned   at  simple  interest  at  6%  to  produce  in  interest 
•jV  of  itself? 

18.  Find  the  exact  interest  of  $1200  for  292  da.  at  5%. 

19.  Find  the  exact  interest  of  $7300  for  146  da.  at  7%. 

20.  The   exact   interest   of    $10,800    at    5%    is   $324. 
Find  the  time. 

21.  In  what  time  will  $260  amount  to  $262.60  at  5%  ? 

22.  What  sum   must  be  deposited  in  a  savings  bank 
which  pays  3-|%  interest  to  produce  semiannually  $8.75? 

23.  A  man  deposits  his  money  in  two  banks.     In  one 
bank  he  has  $572  which  pays  3^%.    The  other  bank  gives 
4%  interest.     If  he  receives  as  interest  the  same  amount 
from  both  banks,  how  much  money  has  he  all  together? 

24.  If  I  invest  half  my  money  at  6%  and  the  remainder 
at  4%,  and  derive  an  income  of  $650  annually,  how  much 
money  have  I  invested  ? 


224  ADVANCED   BOOK  OF  ARITHMETIC 

REVIEW   QUESTIONS 

1.  Define  principal,  rate,  per  cent,  interest,  amount. 

2.  How  does  exact  interest  differ  from  interest  accord- 
ing to  the  common  use  of  the  term?     What  is  the  dis- 
tinction between  simple  interest  and  annual  interest  ? 

3.  How  do  you  find  the  interest  of  a  sum  of  money  at 
a  given  rate  and  for  a  given  time  ? 

4.  If  you  were  given  the  interest,  the  rate,  and  the 
time,  how  would  you  find  the  principal  ? 

5.  Given  the  principal,  interest,  and  time,  how  would 
you  find  the   rate   per   cent?     How  would  you  find  the 
rate  ? 

6.  Given  the  principal,  the  rate,  and  the  interest,  how 
would  you  find  the  time  ? 

7.  Given  the  principal,  the  amount,  and  the  rate,  how 
would  you  find  the  time  ? 

8.  Given  the  principal,  the  amount,  and  the  time,  how 
would  you  find  the  rate  ? 

9.  Given  the  amount,  the  rate,  and  the  time,  how 
would  you  find  the  principal  ? 

10.  If  you  knew  the  interest  of  a  sum  of  money  for  a 
given  time  at  6%,  how  would  you  find  the  interest  for  the 
same  sum  for  the  same  time  at  5%  ?  at  4%  ?  at  8%  ? 

11.  If  you  knew  the  interest  at  4%,  how  would  you  find 
the  interest  of  the  same  sum  at  1%  ?  at  3%  ?  at  5%|?  at 
3i%? 

12.  If  }^ou  were  given  the  interest  of  a  sum  of  money 
for  a  number  of  days,  how  would  you  determine  from  this 
the  exact  interest  of  the  same  sum  for  the  same  number  of 
days? 


PROMISSORY   NOTES  225 

PROMISSORY  NOTES 

A  written  promise  by  one  person  to  pay  another  person 
on  demand,  or  after  a  specified  time,  a  sum  of  money  is 
called  a  promissory  note. 

The  following  are  promissory  notes  written  in  standard 
form : 


Galveston,  Texas,  ?Wcvi&h  7,  1907. 

^t&v  dat&  c/  promise  to  pay  to 


the  order  of 

--&/iis&&  fvwndA&d,  &LqhX/u  ________  —  Dollars 

I         /  100 

at  ______  t/i&  ofi/i&t  ofa£u>w&C  joa/wk  ________________________ 

Value  received,  w-UJ^  imteAs&^t  at  6  %. 


No. 


Dallas,  Texas,  Tn^eA  V,  1907. 
dewuwvcL  c/  promise  to  pay  to  the  order  of 


—Dollars 


at  _____________  tfi 

Value  received,  w£(A  iM,t&^&&t  at  7  %  . 


Due. 


The  first  of  the  above  promissory  notes  is  called  a  time 
note;  the  second  is  called  a  demand  note. 
Q 


226  ADVANCED   BOOK  OF  ARITHMETIC 

The  person  who  promises  to  pay  is  called  the  maker. 
John  Mosley  is  the  maker  of  the  first  note  above. 

The  person  to  whom  the  money  is  to  be  paid  is  called 
the  payee.  The  person  who  has  legal  possession  of  a  note 
is  called  its  holder. 

The  sum  specified  in  a  note  is  called  its  face.  A  time 
note  is  legally  due  on  the  date  indicated.  In  some  states 
3  days  more  than  are  indicated  in  the  note  are  allowed 
before  the  note  is  legally  due.  These  days  are  called 
days  of  grace.  The  day  on  which  a  note  is  legally  due  is 
called  the  day  of  maturity. 

A  note  made  payable  to  the  order  of  a  person,  or  a  note 
made  payable  to  the  bearer  is  negotiable,  i.e.  it  may  be 
transferred  from  one  person  to  another  person. 

A  note  made  payable  to  the  payee  only  is  non-negotiable. 

When  a  note  payable  to  the  order  of  the  payee  is  trans- 
ferred, every  holder  before  parting  with  it  must  indorse  it, 
i.e.  write  his  name  on  the  back  of  it.  Every  indorser 
thus  becomes  liable  for  the  payment  of  the  note,  if  the 
maker  fails  to  pay  it.  The  holder  in  whose  possession 
the  note  is  at  maturity  presents  it  to  the  maker  for  pay- 
ment. If  the  maker  refuses  to  pay  it,  the  holder  engages 
a  Notary  Public  to  give  to  the  indorser,  or  indorsers,  a 
written  notice  of  its  non-payment.  This  notice  is  called 
a  protest.  A  protest  must  be  sent  on  the  date  of  maturity ; 
otherwise  the  indorsers  are  not  held  responsible  for  the 
payment  of  the  note. 

An  indorser  who  writes  over  his  signature  the  words 
ivithout  recourse  is  not  held  responsible  for  the  payment  of 
the  note. 

A  note  made  payable  to  the  bearer  is  negotiable  without 
indorsement.  In  some  states  a  note  must  contain  the 
words  value  received  in  order  to  be  legal. 


BANK  DISCOUNT  227 

If  the  words  with  interest  are  not  in  a  note,  no  interest 
is  charged.  If,  however,  the  note  is  not  paid  on  the  date 
of  maturity,  interest  at  the  legal  rate  may  be  charged.  If 
a  note  contains  the  words  with  interest,  and  no  rate  is  speci- 
fied, it  is  then  understood  that  the  note  bears  the  rate  of 
interest  usually  charged  in  the  state  where  it  is  made. 

When  the  time  of  payment  is  indicated  in  months,  cal- 
endar months  are  understood. 

A  note  drawn  March  6,  and  payable  two  months  after 
date,  matures  on  May  6  in  states  where  days  of  grace  are 
not  allowed,  and  on  May  9  in  states  where  days  of  grace 
are  allowed.  About  one  half  of  the  states  and  territories 
allow  3  days  of  grace. 

BANK  DISCOUNT 


New  Orleans,  La.,  <$c,6..  /P,  1903. 
W.OO. 

c/t/^£y  cLaAf&  a^tsA*  cla,t&  c/  promise  to  pay  to 
the  order  of  ________________  fotefiA,  (^o-an  _________________ 


fauncU&ci  fvfty  __________________________  —Dollars 

Value  received. 

No.  33.      Due  ftfalt  15  1  18, 


The  above  time  note  is  negotiable  when  indorsed. 
Supposing  the  payee,  Joseph  Coan,  needs  money,  he  can 
sell  the  note  to  a  bank.  The  sum  the  bank  gives  him  for 
the  note  is  called  the  proceeds  of  the  note.  The  differ- 
ence between  the  proceeds  and  the  face  of  the  note  is 
called  the  bank  discount. 


228  ADVANCED  BOOK  OF  ARITHMETIC 

The  bank  discount  is  always  a  rate  per  cent  of  the  value 
of  the  note  011  its  day  of  maturity,  reckoned  from  the  date 
of  the  sale  of  the  note  to  the  day  of  maturity. 

The  bank  discount  is  then  the  interest  on  the  maturity 
value  of  the  note  computed  from  the  date  of  discount  to 
the  date  of  maturity.  This  time  is  called  the  term  of 
discount. 

The  maturity  value  of  the  note  minus  the  bank  discount 
is  the  proceeds  of  the  note. 

From  the  computer's  point  of  view  the  essential  features 
of  a  note  are  the  face,  maturity  value,  date  of  drawing,  date 
of  sale,  or  date  of  discount,  rate  of  interest  the  note  bears, 
rate  of  interest  charged,  known  as  rate  of  discount,  and 
date  of  maturity. 

BANKERS'   INTEREST 

Banks  charge  interest  for  the  exact  number  of  days 
between  dates,  allowing  30  days  to  a  month.  Banks 
usually  draw  notes  for  30,  60,  or  90  days. 

Example.  Find  the  discount  and  the  proceeds  of  the 
above  note,  if  it  was  discounted  at  8%,  March  1,  1903. 

SOLUTION.  The  bank  charges  discount  from  March  1, 
to  April  18. 

From  March  1  to  April  18  is  48  days. 

$350.      =  maturity  value. 

.08 

28.00  =  int.  for  1  yr. 

45  da.  =  £  of  1  yr.       3.50  =  int.  for  45  da. 
3  da.  =  T^  of  45  da.     .23  =  int.  for  3  da. 
$3.73  =  int.  for  48  da. 
=  bank  discount. 
$350  -  $3.73  =  $346.27  =  proceeds  of  the  note. 


COMPUTING   DISCOUNT  229 

EXERCISE   123 

Find  the  bank  discount  and  the  proceeds  of  the  follow- 
ing indicated  notes,  allowing  3  days  of  grace  in  examples 
1,  2,  10,  11,  12,  13,  and  no  grace  in  the  remaining:  — 


DATE 

TIME 

FACE 

Dis- 

RATE  OF 

C'TI 

SD 

DISC'T 

1. 

Jan. 

12, 

60 

da., 

$600, 

Feb. 

13, 

8%. 

2. 

July 

4, 

60 

da., 

$800, 

Aug. 

3, 

8%. 

3. 

Mar. 

3, 

90 

da., 

$500, 

Mar. 

3, 

6%. 

4. 

April  5, 

60 

da., 

$700, 

April  5, 

6%. 

5. 

May 

7, 

60 

da., 

$600, 

May 

10, 

10%. 

6. 

May 

9, 

20 

da., 

$900, 

May 

24, 

8%. 

7. 

May 

30, 

90 

da., 

$750, 

July 

1, 

6%. 

8. 

June 

5, 

60 

da., 

$450, 

July 

5, 

6%. 

9. 

Aug. 

11, 

30 

da., 

$800, 

Aug. 

11, 

6%. 

10. 

Sept. 

9, 

30 

da., 

$350, 

Sept. 

12, 

10%. 

11. 

Oct. 

4, 

60 

da., 

$800, 

Oct. 

7, 

10%. 

12. 

Oct. 

14, 

60 

da., 

$500, 

Nov. 

16, 

12%. 

13. 

Nov. 

10, 

90 

da., 

$600, 

Nov. 

25, 

8%. 

14. 

Dec. 

4, 

90 

da., 

$750, 

Feb. 

5, 

6%. 

15. 

Jan. 

10, 

45 

da., 

$650, 

Feb. 

9, 

7%. 

16. 

Jan. 

5, 

60 

da., 

$850, 

Feb. 

5, 

9%. 

17. 

Jan. 

30, 

75 

da., 

$950, 

Feb. 

28, 

9%. 

18. 

July 

10, 

3 

mo., 

$380, 

July 

12, 

9%. 

COMPUTING   DISCOUNT  ON  INTEREST-BEARING  NOTES 

Find  the  bank  discount  and  the  proceeds  of  a  90-day 
note  for  $250,  dated  Portland,  Me.,  June  9,  1907,  bearing 
interest  at  6%,  and  discounted  July  8,  1907,  at  8%. 


230  ADVANCED  BOOK  OF  ARITHMETIC 

Step  1.    Find  the  maturity  value  of  the  note.     Maine 
allows  3  days  of  grace.     Hence,  the  interest  will  be  com- 
puted for  93  days. 
$250 

~      3.87=  interest  at  6%  for  93  days. 
$253.87  =  maturity  value  of  the  note. 
Step  2.    Find  the  term  of  discount  (exact  number  of  days 
from  July  8,  to  Sept.  10,  the  date  of  maturity). 

Step  3.  Find  the  bank  discount.  This  is  reckoned  on 
the  maturity  value  of  the  note. 

Int.  on  $253.87  for  64  days  at  8%  =$3. 61. 
$254.87  -  $  3.61  =  $250.26,  proceeds  of  note. 

EXERCISE  124 

Find  the  discount  and  the  proceeds  of  the  following  in- 
dicated notes,  allowing  3  days  of  grace  in  examples  2,  3, 
6,  9,  10,  13,  and  no  grace  in  the  remaining  :  — 


FACE 

DATE 

TIME 

J.W1.J.  K, 

OF  INT. 

J.V.A.  J.  Hi   \J 

DISC'T 

DISC'T 

1. 

$350, 

Jan. 

1, 

45 

da., 

10%, 

12%, 

Feb. 

1. 

2. 

$395, 

Jan. 

10, 

3 

mo., 

6%, 

10%, 

Jan. 

20. 

3. 

$450, 

Feb. 

1, 

30 

da., 

6%, 

8%, 

Feb. 

18. 

4. 

$600, 

Mar. 

1, 

60 

da., 

8%, 

12%, 

Mar. 

31. 

5. 

$500, 

Apr. 

2, 

60 

da., 

6%, 

8%, 

Apr. 

17. 

6. 

$900, 

Apr. 

10, 

30 

da., 

6%, 

8%, 

Apr. 

10. 

7. 

$1000, 

Apr. 

15, 

60 

da., 

8%, 

10%, 

Apr. 

15. 

8. 

$750, 

May 

4, 

60 

da., 

6%, 

6%, 

May 

5. 

9. 

$800, 

July 

10, 

30 

da., 

6%, 

8%, 

July 

26. 

10. 

$400, 

Aug. 

15, 

60 

da., 

6%, 

6%, 

Aug. 

15. 

11. 

$850, 

Nov. 

11, 

90 

da., 

1%, 

10%, 

Nov. 

12. 

12. 

$900, 

Dec. 

12, 

45 

da., 

6%, 

6%, 

Jan. 

13. 

13. 

$1200, 

Dec. 

5, 

3 

mo., 

6%, 

10%, 

Jan. 

15. 

COMMERCIAL  DISCOUNTS  231 

COMMERCIAL  DISCOUNTS 

Commercial,  or  Trade  Discount  is  a  reduction  from  the  list 
price  of  goods,  or  the  amount  of  a  bill. 

If  two  or  more  discounts  are  allowed,  the  first  is 
reckoned  on  the  list  price,  the  next  on  the  remainder  after 
deducting  the  first  discount,  the  third  is  reckoned  on  the 
second  remainder,  etc. 

Example.  Find  the  cost  price  of  a  bill  of  goods,  if  the 
list  price  is  $690  and  discounts  of  25%,  10%,  and  5%  are 
allowed. 

SOLUTION.     $690 

.$172.50    =     25%  of  $600. 
$517.50    .=     first  remainder. 
$  51.75    =     10%  of  $517.50. 
$465.75    =     second  remainder. 
$  23.287  =     5%  of  $465.75. 
$442.46    =   cost  of  the  goods. 

EXERCISE   125 

1.  Find    the   cost    if    the  list    price    is  $350,  and  the 
discount  20%. 

2.  Find  the  cost  when  the  list  price  is  $823,  and  the 
discount  12|%. 

3.  What  is  the  cash  value  of  a  bill  of  goods  listed  at 
$937.50  when  a  discount  of  16|%   is  given? 

4.  A  suit  of  clothes  is  marked  $17.50,  and  is  sold  for 
$12.00.     What  is  the  rate  of  discount  ? 

5.  A  bookseller  buys  60  books  marked  $2.10  each  at  a 
discount  of  16|%,  and  sells  them  at  the  marked  price. 
What  is  his  gain  per  cent,  and  how  much  profit  does  he 
make  on  the  sale  of  the  books? 


232  ADVANCED  BOOK  OF  ARITHMETIC 

6.  A  dealer  buys  goods  at  a  discount  of  16|  %  and  sells 
them  at  5%  above  the  list  price.     What  is  his  gain  per 
cent? 

7.  Find  the  cost  price  in  each  case,  if  the   list  prices 
and  rates  of  discount  are  as  follows  : 

COST  DISCOUNTS 

(a)  $700  20%,  121%,  10% 

(5)  $7000  10%,  20%,  30% 

(<?)  $3000  15%,  12% 

(rf)  19690  |%,33i% 

(e)  $5000  43%,  10% 

(/)  $3000  30%,  20%,  20% 

(#)  $  4000  20%,  10%,  5% 

Qi)  $6000  46%,  45% 

(l)  $2760  30%,  15% 

PARTIAL    PAYMENTS 

When  payments  are  made  on  a  note,  these  payments 
are  known  as  partial  payments.  These  payments  and  their 
dates  of  payment  are  written  on  the  back  of  the  note. 

There  are  several  methods  of  computing  the  amounts 
due  on  such  notes.  The  best  known  and  most  widely 
used  are  the  United  States  Rule  and  the  Merchants'  Rule. 

UNITED  STATES  RULE 

Find  the  amount  of  the  principal  until  the  time  of  the  first 
payment,  or  until  the  sum  of  two  or  more  payments  equals  or 
exceeds  the  interest. 

Subtract  the  payment,  or  the  sum  of  the  two  or  more 
payments  from  the  amount. 

Proceed  with  the  remainder  as  a  new  principal.  Continue 
in  this  manner  until  the  date  of  settlement. 


PARTIAL  PAYMENTS 


233 


A    note     for    $600.00    dated    Aug.     10,    1902,    was 
indorsed  as  follows  : 

1902,  Dec.  15,  $100. 

1903,  Feb.  12,  $150. 
1903,  March  15,  $150. 

Find  the  amount  due  April  1,  1903. 


SOLUTION. 

DATES  OF  INDORSEMENT 
yr.  mo.  da. 

1902  8  10 
1902  12  15 
1903 
1903 


TIME  BETWEEN  DATES 
yr.  mo.  da. 

4          5 


2 
3 
4 


12 
15 
1 


27 

3 

16 


PAYMENTS 
$100 

$150 
$150 


1903 

$600       =  first  principal,  $600  x  .06  x  -|  =  $12.50. 

12.50  =  int.  for  4  mo.  5  da. 
$612.50  =  amt.  for  4  mo.  5  da. 
$100       =  first  payment.  _,  Q 

$512.50  =  second  principal.    $512.50  x.06  x  -^  =  $4.87. 

4. 87  =  int.  for  1  mo.  27  da. 
$517.37  =  amt.  for  1  mo.  27  da. 

150.      =  second  payment.  ^  _, 

$367.37  =  third  principal.     $367.37  x  .06  x  ^  =  $2.02. 

2.02  =  int.  for  1  mo.  3  da. 
$369.39  =  amt,  for  1  mo.  3  da. 

150.      =  third  payment.  ^  „ 

$219. 39  =  fourth  principal.   $219.39  x  .06  x^  =  $-59. 

.59  =  int.  for  16  da. 
$219.98  =  amt.  for  16  da.      $219.98.  Am. 

NOTE.  Problems  of  this  type  are  not  as  common  in  business  as  they 
were  some  years  ago.  Payments  are  now  usually  made  at  equal  intervals 
of  time. 


234  ADVANCED  BOOK  OF  ARITHMETIC 

EXERCISE   126 

1.  A  man  borrows  from  a  loan  association  $3000  at 
8%  interest.     If  he  pays  $  100  at  the  end  of  every  3  mo. 
for  1  yr.,  find  the  amount  due  at  the  end  of  the  year. 

2.  A  man  borrows  from  a  building  and  loan  company 
$2000  at  8%  interest.     He  pays  in  monthly  installments 
of  $50  each.     How  much  does  he  owe  at  the  end  of  6  mo.  ? 

3.  If   you    borrow  $1800    at    6  %    interest   and    pay 
annually  $  600,  how    much  do  you   owe  at  the    end   of 
3yr.? 

4.  If  you  borrow  $1500  at  8%  interest  and  pay  semi- 
annually  $300,   how   much    do  you  ..owe  at  the   end  of 
2yr.?   " 

5.  If  $2400  is  borrowed  at  8%  interest,  and  paid  in 
quarterly  installments  of  $150  each,  how  much  is  due  at 
the  end  of  two  years  ? 

6.  A  man  borrows  $  3000  at  8  %  interest,  and  pays  in 
semiannual  installments  of  $500  each.     How  much  is  due 
at  the  end  of  three  years  ? 

7.  A  man  borrows  $1800  at  6%,  and  pays  $400  per 
year.     How  much  is  due  at  the  end  of  five  years  ? 

8.  If  I  borrow  $2500  at  5%  interest,  and  pay  $500  a 
year,  how  much  do  I  owe  at  the  end  of  five  years? 

9.  How  much  is  due  at  the  end  of  two  years  on  a  loan 
of  $2500  at  5  %  interest  when  paid  in  semiannual  install- 
ments of  $600  each? 

10.  A  note  dated  Jan.  15,  1904,  for  $4000  had  the  fol- 
lowing indorsements  :  July  10,  1904,  $700  ;  Dec.  24, 1904, 
$600;  June  18,  1905,  $800;  Nov.  13,  1905,  $500;  March 
17,  1906,  $900;  Aug.  10,  1906,  $400; /Feb.  12,  1907, 
$400.  How  much  was  due  May  28,  1907,  at  6%? 


PARTIAL  PAYMENTS  235 

THE   MERCHANTS'   RULE 

The  Merchants'  Rule  for  computing  interest  on  partial 
payments  is  used  when  settlement  is  made  within  one 
year  from  the  date  of  the  note.  The  rule  is  as  follows  : 

I.  Compute  the  amount  of  the  face  of  the  note  until  the 
date  of  settlement. 

II.  Compute  the  amount  of  each  indorsement  from  its  date 
until  the  date  of  settlement,  and  add  these  amounts. 

III.  Subtract  the  sum  of  the  amounts  of  the  indorsements 
from  the  amount  of  the  face  of  the  note.      The  remainder -will 
be  the  amount  due  on  the  date  of  settlement. 

Example.  A  note  for  $500,  with  interest  at  8%,  dated 
Jan.  10,  1902,  had  the  following  indorsements  :  April  1, 
1902,  $100;  May  10,  1902,  $200;  July  1,  1902,  $100. 
Find  the  amount  due  Nov.  1,  1902. 

SOLUTION.    From  Jan.  10  to  Nov.  1  is  9  mo.  21  da. 

From  April  1  to  Nov.  1  is  7  mo. 

From  May  10  to  Nov.  1  is  5  mo.  21  da. 

From  July  1  to  Nov.  1  is  4  mo. 

$500  for  9  mo.  21  da.  amounts  to  $532.33 

$100  for  7  ino.  amounts  to  $104.67 

$200  for  5  mo.  21  da.  amounts  to       207.60 

$100  for  4  mo.  amounts  to  102.67 

The  payments  amount  to  414.94 

Balance  due  Nov.  1,  $117.39 

EXERCISE  127 

l.  A  note  for  $500,  with  interest  at  6  %,  dated  Jan.  2, 
1902,  had  the  following  indorsements  :  March  2,  1902, 
$100;  May  5,  1902,  $150;  July  10,  1902,  $200.  Find 
the  amount  due  Nov.  1,  1902. 


236  ADVANCED  BOOK  OF  ARITHMETIC 

2.  A  note  for  $600,  with  interest  at  8%,  dated  Feb.  1, 
1902,    was  indorsed  as  follows:    March   11,    1902,  $200; 
May  15,  1902,  $100;  July  3,  1902,  $100.     Find  the  bal- 
ance due  Aug.  1,  1902. 

3.  A  note  for  $800,  with  interest  at  8%,  dated  Feb.  10, 
1902,  was   indorsed  as  follows;  March   15,   1902,  $200; 
April   10,   1902,  $100;    June   3,  1902,  $200.     Find    the 
amount  due  Nov.  1,  1902. 

4.  A  borrows  $1500  on  Jan.  1,  1907,  at  6%  interest, 
and  pays  $450  each  quarter.     How  much  does  he  owe 
when  three  payments  are  made  ? 

EXCHANGE 

The  written  order  of  one  party  to  another,  requesting 
the  payment  of  a  specified  sum  of  money,  is  called  a  draft. 
The  parties  to  a  draft  are  drawer,  drawee,  and  payee; 
The  following  are  forms  of  drafts: 

SIGHT  DRAFT 


New  Orleans,  La.,  mwieh  fO,  1903. 
/  600.00 

,  pay  to  the  order  of  ___________________ 

jla&ok  s'i&cf&t  ____________  the  sum  of 


Dollars. 


Value  received  and  charge  to  the  account  of 

TO 


EXCHANGE  237 

A  time  draft  payable  after  sight  matures  as  a  promis- 
sory note. 

Bank  drafts  form  one  of  the  most  important  mediums 
of  exchange.  If  a  merchant  owes  money  in  New  York, 
Chicago,  or  in  any  other  place,  he  can  always  purchase  a 
draft  from  his  home  bank,  and  by  transmitting  this  through 
the  mails  settle  his  indebtedness. 

TIME  DRAFT 


Louisville,  Ky.,  fl/ay.  /O,  1902. 
$1500.00 

da,t&  pay  to  the  order  of 


_  .<3:vft&&yv  Aat  ndb&d .Q  Dollars. 


Value  received  and  charge  to  the  account  of 


To 

No.  8.  TniwttaotU,  Tninn.  tfet. 


BANK   DRAFT 


Galveston,  Texas,  7MaA*A,  10,  1903. 
No.  7^. 

f^utcfjmp,  ^ealg  ^  Co.,  Bankers, 
Pay  to  the  order  of.  ______ 


f  300.00  ___________  (&6AM  kwvicii/£cC)  __________  -^Dollars. 


-^ 
Cashier. 


To 


238  ADVANCED  BOOK  OF  ARITHMETIC 

The  price  of  exchange  is  a  matter  of  supply  and  de- 
mand. If,  for  instance,  the  Galveston  banks  freely  offer 
New  York  or  Chicago  exchange,  a  purchaser  will  very 
probably  get  it  for  less  than  its  face  value.  If  exchange 
is  scarce,  its  price  is  high.  When  exchange  costs  more 
than  its  face  value,  it  is  said  to  be  at  a  premium  ;  when  it 
costs  less  than  its  face  value,  it  is  said  to  be  at  a  discount. 
Exchange  is  at  par  when  the  cost  of  a  draft  is  its  face 
value.  Exchange  is  always  quoted  as  a  rate  per  cent, 
or  as  so  many  dollars  on  a  $1000.  The  exchange  is 
always  reckoned  on  the  face  value  of  the  draft.  Thus 
\c/0  premium,  or  $1.25  premium,  means  that  the  cost  of  a 
draft  for  $100  is  $100|,  and  the  cost  of  a  draft  for  $1000 
is  $1001.25.  The  rate  of  exchange  is  generally  a  fraction 
of  1%. 

Express  companies  charge  for  money  orders  to  any  part 
of  this  country  or  Canada  not  over  30^  for  $100.  It  is 
reasonable  to  assume  that  a  person  will  not  pay  much 
more  than  this  rate  for  exchange. 

EXERCISE  128 

1.  What  is  the  cost  of  exchange  on  a  sight  draft  for 
$400  at  1%  premium? 

2.  What  is  the  cost  of  exchange  on  a  sight  draft  for 
$300  at  |%  premium? 

3.  Find  the  cost  of  a  draft  on  New  York  for  $200  at 
\°/0  premium. 

4.  Find  the  cost  of  a  demand  draft  for  $500  at    \% 
premium. 

5.  Find   the  cost  of  a  sight   draft  for  $600   at 
premium. 


EXCHANGE  239 

6.  Find  how  much  must  be  paid  for  a  sight  draft  for 
$1000  at  $1.25  premium. 

7.  Find  how  much  must  be  paid  for  a  sight  draft  for 
12000  at  $2.50  premium. 

8.  Find    the  cost  of  remitting  $800    from  Dallas  to 
Chicago  by  means  of  a  bank  draft  when  exchange  is  ^  % 
premium. 

9.  Find  the  cost  of  a  sight  draft  on  Chicago  for  $700, 
exchange  being  at  |  %  discount. 

10.  Find  the  cost  of  a  sight  draft  on  New  York  for 
$400  when  exchange  is  -|%  discount. 

11.  Find  the  cost  of  exchange  on  a  draft  for  $1000, 
exchange  being  |  %  premium. 

12.  Find  the  cost  of  an  express  order  for  $1200  at  the 
rate  of  30^  per  $100. 

13.  If  you  had  to  remit  $200  to  Chicago  when  exchange 
is  \  %  premium,  which  would  you  prefer,  to  buy  exchange 
or  to  buy  an  express  money  order  at  the  rate  of  30^  for 
$100? 

14.  What  is  the  cost  of  an  express  money  order  for 
$1000? 

15.  What    is  the  cost   of   a  sight  draft  for   $3000   at 
$1.50  premium  ? 

16.  What  is  the  cost  of  a  sight  draft  for  $3000  at  $2.50 
premium  ? 

Example.     What  is  the  cost  of  a  draft  on  Buffalo  for 
$800,  payable  in  30  da.  at  6%    interest,  exchange  being 
at  the  rate  of  ^  %  premium  ? 
SOLUTION  : 

The  interest  on  $800  for  30  da.  at  6%  =$4.00. 
Cost  of  exchange  =  ^  %  of  $800  =  $2.00. 
The  cost  of  the;  draft .=  $800  4-  $2  -  $4  =  $798.        /i 


240 


ADVANCED  BOOK  OF  ARITHMETIC 


EXERCISE   129 

Find  the  cost  of  the  following  sight  drafts : 


l. 
2. 


FACE 

$2720, 
$3480, 


3.  $5080, 

4.  $6290, 

5.  $8290, 
$9980, 
$5493, 

8.  $5280, 

9.  $6040, 
10.  $6090, 

$5400, 
$9870, 
$4500, 
$6920, 
$7780, 
16.  $1234, 


6. 
7. 


11. 
12. 
is. 
14. 
is. 


RATE  OF  EXCHANGE 
\%  premium. 
i%  premium. 
^°fo  premium. 
$2.50  premium. 
$1.25  discount. 
\%  discount. 
$1.50  discount. 
$1.50  premium. 
$2.00  discount. 
$1.25  premium. 
$1.25  discount. 
1%  discount. 
\%  premium. 
discount. 


\°/o  premium. 
\°/o  discount. 


17.  Find  the  cost  of  an  express  money  order  for  $1400 
at  30^  for  $100. 

18.  Find  the  cost  of  a  draft  for  $10,000  at  \%  discount. 

19.  Find  the  cost  of  a  draft  for  $600  payable  in  30  da. 
without  grace,  the  rate  of  interest  being  6%,  exchange 
being  |%  premium. 

20.  Find  the  cost  of  a  draft  for  $  1000  payable  in  30  da., 
exchange  being  at  par  and  the  rate  of  interest  being  &%. 

21.  Find  the  cost  of  a  45  da.  draft  for  $  900,  exchange 
being  at  \%  discount,  and  the  rate  of  interest  being  6%. 


EXCHANGE 


241 


VALUE  OF  FOREIGN  COINS  IN  UNITED  STATES  MONEY 

(Proclaimed  by  the  Secretary  of  the  Treasury  Oct.  1,  1906) 


COUNTRY 

STANDARD 

MONETARY  UNIT 

VALUE  IN 
U.  S.  GOLD 
DOLLAR 

Argentine  Republic     .     . 
Austria-Hungary    .     .     . 
Belgium          

Gold 
Gold 
Gold 
Silver 
Gold 
Gold 
Silver 
Gold 

Silver 

Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 
Gold 

Peso 
Crown 
Franc 
Boliviano 
Milreis 
Dollar 
Peso 
Peso 
(  Shanghai 
Tael       <  Haikwan 
(  Canton 
Dollar 
Colon 
Crown 
Sucre 
Pound  (100  piasters) 
Franc 
Mark 
Pound  sterling 
Drachma 
Gourde 
Pound  Sterling 
Lira 
Yen 
Peso 
Florin 
Dollar 
Crown 
Balboa 
Libra 
Milreis 
Ruble 
Peseta 
Crown 
Franc 
Piaster 
Peso 
Bolivar 

$  0.965 
.203 
.193  *. 
.485 
.54(5 
1.00 
.485 
.365 
.726 
.808 
.792 
1.00 
.465 
.268 
.487 
4.943 
.193 
.238 
4.8665 
.193 
.965 
4.8665 
.193 
.498 
.498 
.402 
1.014 
.268 
1.000 
4.8665 
1.08 
.515 
.193 
.268 
.193 
.044 
1.034 
.193 

Bolivia        

Brazil     

Canada  

Central  America     .     .     . 
Chile  

China             

Ecuador 

E°°VDt 

France             

Germany    

Great  Britain      .... 
Greece   

Haiti      

India      
Italy  

Japan          .     . 

IVIexico        .     .          .     . 

Netherlands   .... 

Newfoundland    .... 
Norway      

Panama      

Peru  

Portugal 

Russia    .     .              . 

"•"•""Spain     

Sweden  

Turkey  

Uruguay    

The  Circulars  of  the  Secretary  of  the  U.  S.  Treasury, 
issued  every  3  months,  fix  the  legal  equivalent  of  the 
monetary  units  in  the  principal  countries.  In  countries 
which  have  adopted  the  silver  standard  monetary  units 
fluctuate  in  value,  owing  to  changes  in  the  market  value 
of  silver. 


242  ADVANCED  BOOK  OF  ARITHMETIC 

Example  l.    Express  $100  in  florins  (Netherlands). 
$  .402  =  1  florin. 


...  $  100  -  -        of  1  florin  =  248.76  florins. 


EXERCISE   130 

1.  Express  the  value  of  <£!  in  French  currency;   in 
German  currency  ;  in  Austrian  currency. 

2.  Change  to  United  States  currency  (a)  450  francs, 
(6)  550  pesetas,  (c)  280  Norwegian  crowns,  (ci)  900  marks, 
(e)  200  colons,  (/)  720  Austrian  crowns,  (#)  450  Danish 
crowns. 

3.  Find  in  United  States  currency  the  difference  in 
value  between  1000  francs  and  f  198. 

4.  Find  the  value  of  $1000  (a)  in  the  money  of  Costa 
Rica;   (6)  in  Austrian  currency;   (<?)  in  Swedish  currenc}7". 

5.  What  is  the  equivalent  in  United  States  money  of 
200,000  bolivars  (Venezuela)  ? 

6.  What  is  the  equivalent  in  United  States  money  of 
10,000  Mexican  pesos  ? 

7.  Express  $1  in  Peruvian  monetary  units. 

8.  Express  $1  in  Norwegian  monetary  units. 

9.  Find  in  United  States  currency  the  difference  be- 
tween the  value  of  10,000  Japanese  yen  and  $5000. 

10.    What  is  the  value  of  10,000  Portuguese  milreis? 

ENGLISH  MONEY 

4  farthings  =  1  penny  (c?.) 
12  pence        =  1  shilling  (.s.) 
20  shillings  =  1  pound  sterling  (£) 
£1  =  20s.  =  240(7.  =  960  farthings 

The  abbreviations  «£,«.,  c?.,  are  the  initial  letters  of  the 
names  of  the  Roman  coins,  libra,  solidus,  denarius. 


EXCHANGE  243 

EXERCISE  131 

1.  What  is  the  equivalent  of    7s.   5|c?.  in  U.   S.   cur- 
rency ? 

7s.  5jd.  =  (l  +  — V  =  7.479166s.  =  £  .373958. 
V         12  / 

$4.8665  x  .37396  =  11.82. 

2.  The  average  freight  rate  on  wheat  per  bushel  for 
the  year  1906  from  Chicago  to  New  York  was : 

By  lake  and  canal  5.94^. 

By  lake  and  rail  6.48^. 

By  all  rail  delivered  to  steamer  8.10/,  and  from  New 
York  to  Liverpool  l^gC?. 

Find  the  cost  of  shipping  5  carloads  of  wheat  averaging 
79,000  Ib.  from  Chicago  to  Liverpool. 

(a)  By  lake  and  canal  to  seaboard  and  thence  by 
steamer. 

(5)  By  lake  and  rail  to  seaboard  and  thence  by  steamer. 

(<?)  By  rail  to  seaboard  and  thence  by  steamer. 

3.  Find  the  cost  of  shipping  from  New  York  to  Liver- 
pool 12,000  bu.  of  wheat  at  l-f^d.  per  bu. 

4.  Find  the  cost  of  shipping  50  tons  of   sacked  flour 
from  Chicago  to  Liverpool  at  lO.llcZ.  per  100  Ib. 

5.  Find  the  cost  of  shipping  from  Galveston  to  Liver- 
pool :  — 

(a)   10,884,000  Ib.  of  cotton  at  Is.  6%d.  per  100  Ib. 
(5)    19,490  bu.  of  wheat  at  M.  per  bu. 
((?)    37,920  bu.  of  corn  at  5d.  per  bu. 

6.  Find  the  cost  of  shipping  from  St.  Louis  to  Liver- 
pool via  New  Orleans  :  — 

(a)   6475  bbl.  of  flour  at  Is.  8%d. 

(>)    97330  bu.  wheat  at  7fd. 

In  examples  5  and  6  take  £  1  =  $4.80. 


244  ADVANCED  BOOK  Ol?  ARITHMETIC 


FOREIGN  EXCHANGE 

The  settling  of  outstanding  indebtedness  between  par- 
ties in  different  parts  of  this  country  by  means  of  drafts 
is  known  as  domestic  exchange.  The  settling  of  debts 
between  parties  in  this  country  and  parties  in  foreign 
countries  by  means  of  drafts  is  called  foreign  exchange. 
Foreign  drafts  are  known  as  bills  of  exchange.  These  are 
issued  in  duplicate,  i.e.  two  are  drawn  exactly  alike.  The 
two  bills  are  sent  by  different  mails.  As  soon  as  one  is 
paid,  the  other  is  void. 

Usually  exchange  is  a  little  above  or  a  little  below  the 
par  of  exchange,  i.e.  intrinsic  value  of  foreign  coins. 

Exchange  on  Great  Britain  is  quoted  at  the  number  of 
dollars  to  <£!;  exchange  on  France  is  quoted  at  so  many 
cents  to  the  franc,  or  as  so  many  francs  to  the  dollar ;  ex- 
change on  Germany  is  quoted  as  so  many  cents  to  4 
marks,  or  as  so  many  cents  to  a  mark. 

BILL  OF  EXCHANGE 


No.  24-89  6  New  York,  N.Y.,  fan.  V-,  1903. 


At  sight  of  this  first  of  exchange  (second  of 

the  same  date  and  tenor  unpaid)  _______________________ 

________________________________________________  .pay  to  the 

order  of.  ____________  ,  ____  1 


Value  received  and  ^arge  the  same  to  the  account  of 
To 


FOREIGN   EXCHANGE  245 

EXERCISE  132 

1.  Find  the  cost  of  a  draft  on  London  for  £1000,  ex- 
change being  4.872. 

2.  Find  the  cost  of  a  bill  of  exchange  for  £550  15s., 
exchange  being  4.85. 

3.  Find  the  cost  of  a  bill  of  exchange  for  5750  francs, 
exchange  being  5.15,  i.e.  5.15  francs  =  $1. 

4.  Find  the  cost  of  a  draft  on  Manchester  for  £3500 
6s.  Sd.  when  exchange  is  quoted  at  4.87J. 

5.  Find  the  cost  of  a  draft  on  Berlin  for  1480  marks 
when  exchange  is  quoted  at  95|,  i.e.  4  marks  =  95  J^. 

6.  Find  the  cost  of  a  draft  on  Vienna  for  8500  crowns, 
rate  of  exchange  being  1  crown  =  $.202. 

7.  Find  the  cost  of  a  draft  for  1500  lira  when  exchange 
is  quoted  at  19.3,  i.e  19.  3  /  =  1  lira. 

8.  Find  the  cost  of  each  of  the  following  drafts  : 

FACE  WHERE  PAYABLE  RATE  OF  EXCHANGE 

(a)   £928  10s.  Liverpool  4.85 

(6)   £  850  6s.  8  d.  Belfast  4.86J 

O)    2500  marks  Berlin  24 

(d)  9280  francs  Havre  5.16 

(e)  8500  lira  Naples  19  J 
(/)  £584  13s.  4<Z.  Dublin  4.85f 

Example.  What  is  the  face  of  a  New  York  draft  on 
Liverpool,  which  costs  $7297.  50,  when  exchange  is  quoted 
at  4.  861-  ? 

SOLUTION.     $4.865  =  £  1. 

£l 


17297.50  =  :     =  £  1500. 


246  ADVANCED  BOOK  OF  ARITHMETIC 

EXERCISE   133 

Find  the  face  of  each  of  the  following  drafts : 

COST  WHERE  PAYABLE  RATE  OF  EXCHANGE 

1.  $5835  London  4.86J 

2.  $1218.75  Newcastle  4.87J 

3.  19063.05  Berlin  95.2 

4.  13724.90  Paris  19.3 

5.  1718.24  Christiana  26.8 

6.  $2366.82  Belfast  4.86J 

7.  $1037  Paris  5.18J 

8.  $5664.80  Hamburg  23.8 

STOCKS  AND  BONDS 

A  company  or  corporation  is  a  number  of  persons  asso- 
ciated under  a  state  law  for  the  purpose  of  transacting 
business. 

The  modern  company,  or  corporation,  has  supplanted 
in  a  great  degree  the  old-time  partnership.  Nowadays 
business  is  conducted  on  so  extensive  a  scale  that  a  com- 
pany often  numbers  several  thousand  persons. 

The  money  which  a  company  invests  in  business  is 
called  capital,  or  stock. 

The  stock  of  a  company  is  usually  divided  into  shares 
of  $  10,  $  50,  or  $  100  each.  The  face  value  of  a  share  in 
most  large  companies  is  $  100. 

These  shares,  as  a  rule,  can  be  bought  and  sold.  The 
buying  and  selling  is  usually  done  by  agents  called  stock- 
brokers. The  usual  fee  of  a  stockbroker  is  at  the  rate  of 
l  %  of  the  par  value  for  buying  a  share  and  \  %  of  the 


STOCKS  AND   BONDS  247 

par  value  for  selling  a  share.  A  broker's  fee  is  called 
brokerage. 

If  the  business  of  a  company  is  prosperous,  a  share  may 
sell  for  more  than  its  face  value.  The  stock  is  then  said 
to  be  at  a  premium.  If  the  business  is  not  prosperous, 
the  value  of  a  share  in  the  market  is  likely  to  be  less 
than  its  face  value.  The  stock  is  then  said  to  be  at  a 
discount. 

When  a  share  sells  in  the  market  for  its  face  value,  it  is 
said  to  be  at  par.  A  quotation  "  10  %  premium  "  means 
the  same  as  110%  of  the  face  value.  A  quotation  "10% 
discount "  means  the  same  as  90  %  of  the  face  value. 

The  profits  of  a  company,  usually  distributed  to  stock- 
holders annually  or  semiannually,  are  called  dividends. 
The  dividend  is  generally  so  many  dollars  on  a  share, 
or  a  percentage  of  the  face  value  of  the  stock.  Thus, 
"4%  dividend"  means  f  4  on  a  share  whose  face  value 
is  $100. 

Stock  is  of  two  kinds,  common  and  preferred.  The  pre- 
ferred stock  is  guaranteed  a  specified  dividend  whether  the 
business  is  profitable  or  not.  The  common  stock  earns 
what  is  left  over  after  expenses  and  the  dividends  on  the 
preferred  stock  are  paid. 

A  bond  is  a  written  instrument  made  by  a  government, 
municipal,  state,  or  national,  or  by  a  corporation,  for  the 
purpose  of  borrowing  money. 

Bonds  are  of  two  kinds,  registered  and  coupon.  A 
registered  bond  is  one  which  is  recorded  by  number  and 
by  name  of  the  owner,  and  it  is  not  transferable  except  in 
writing  and  at  the  office  of  the  treasurer.  Coupon  bonds 
are  so  called  because  they  have  interest  slips  attached  to 
them,  which  are  cut  off  as  the  interest  falls  due.  These 
interest  slips  are  payable  to  the  bearer. 


248 


ADVANCED   BOOK  tfF  ARITHMETIC 


The  following  is  quoted  from  a  daily  newspaper 
the  New  York  stock  market  March  11,  1903. 

Stocks 

giving 
CLOSING 

Atchison     .... 

SALES 

30,200 

HIGHEST 

82 
971 
93 

LOWEST 

801 
97i 

BID 

974 

o 

92$ 

Atchison  pf.    . 

1,800 

Baltimore  &  Ohio     . 

.     14,400 

Canadian  Pacific 

.     13,800 

1291 

127| 

1291 

Manhattan  L  . 

.     17,300 

142| 

141f 

142f 

Metropolitan  St.  Ry. 

.     21,100 

1351 

131f 

134 

Missouri  Pacific  . 

.     30,800 

1081 

106$ 

108f 

Pennsylvania  . 
St.  Paul      .... 

.     53,400 

,     58,500 
.     54,400 

144| 
168$ 

1421 
1671 
61i 

144 
168J 
62$ 

Southern  Pacific  . 

Bonds 

U.  S.  new  4s  registered,  135  U.  S.  old  4s  registered,  103 
U.  S.  new  4s  coupon,  136  Baltimore  &  Ohio  4s,  102|- 
U.  S.  old  4s  coupon,  109J-  Chicago  B.  &  Q.  new  4s,  9S| 
Notice  that  in  stock  quotations  the  fractions  used  are 
halves,  quarters,  eighths.  Notice  further  that  the  varia- 
tions in  the  prices  of  stock  in  the  course  of  a  day  are 
quite  considerable.  There  are  instances  where  stocks 
have  fallen  in  one  day  50%  of  their  face  value.  Why  the 
prices  vary  so  much  would  take  the  proverbial  Phila- 
delphia lawyer  to  explain.  In  the  above  quotations  the 
first  column  shows  the  number  of  shares  sold ;  the  second 
column  gives  the  highest  prices  paid  for  the  shares;  the 
third  gives  the  lowest  prices  paid ;  and  the  fourth  gives 
the  closing  bids.  In  the  course  of  the  day  above  referred 
to  there  were  several  other  prices  than  those  given. 


STOCKS  AND  BONDS  249 

Example.     Find  the  cost  of  10  shares  of  stock  at  107 J. 

SOLUTION 

Market  price         =  $107|. 
Brokerage  =$|. 

Cost  of  1  share     =  |107f  (i.e.  $107J  +  f  J). 
Cost  of  10  shares  =  $107*  x  10  =  $1076.25. 

o 

EXERCISE   134 

1.  Give  the  premium  or  discount  of  each  of  the  stocks 
quoted  March  11,  1903. 

2.  What  is  the  cost  of  20  shares  of  Atchison  at  80  J  ? 
at  81£? 

3.  What  is  the  cost  of  40  shares  of  Baltimore  &  Ohio 
at  92|  ? 

4.  What  is  the  cost  of  200  shares  of  Canadian  Pacific 
at  145 J ? 

5.  What  is  the  cost  of  25  shares  of  stock  at  118|  ? 

6.  Find  the  cost  of  50  shares  of  Missouri  Pacific  at 
106f. 

7.  Find  the  cost  of  100  shares  of  each  of  the  stocks 
quoted  March  11,  1903,  at  the  prices  indicated  in   the 
fourth  column. 

Example  1.    What  sum  should  be  received  from  the  sale 
of  40  shares  of  Pennsylvania  at 

SOLUTION 
Market  value  of  1  share  of  stock  = 

Brokerage  =  §\. 

Sum  received  from  1  share  =  $144. 
Sum  received  from  40  shares  =  1144  x  40  =  15760. 


250  ADVANCED  BOOK  OF  ARITHMETIC 

Example  2.    What  profit  is   made    by    purchasing    50 
shares  of  stock  at  142  J  and  selling  them  at  $  144  ? 
SOLUTION.     Cost  of  1  share  =  $1421  +  $£  =  $  142f. 
Receipts  from  1  share  =  $144  —  $£  =  (J143J. 

Gain  on  1  share  =  $  1^. 
Gain  on  50  shares  =  $1 J  x  50=$ 75. 

EXERCISE   135 

1.  What  should  a  person  receive  from  the  sale  of  100 
shares  of   Manhattan   at   the    closing  bid  above  given  ? 
from  the  sale  of  100  shares  at  the  lowest  bid  ? 

2.  How  much  should  a  person  receive  from  the  sale 
of  100  shares  of  Atchison  preferred  at  97 J  ? 

3.  How  many  shares  of  Pennsylvania  at  142^  can  be 
bought  for  $14,237.50? 

4.  What  profit  is  made  by  buying  100  shares  of  Bal- 
timore &  Ohio  at  91 J-,  and  selling  them  at  92|-? 

5.  If  100  shares  of  St.  Paul  are  bought  at  167|  and 
sold  the  same  day  at  168-|,  what  is  the  gain? 

6.  If  200  shares  of  New  York  Central  are  bought  at 
165  and  sold  the  following  day  at  167^,  what  is  the  gain? 

7.  A  speculator  bought  Missouri  Pacific  at  the  lowest 
price   quoted   March   11,  1903,  and   sold   at   the   closing 
bid  of  the  same  day.     If  his  gain  was  $300,  how  many 
shares  did  he  buy  ? 

8.  If  a  speculator  buys  Metropolitan  Street  Railway 
at  131 1  and  sells  on  the  same  day  at  134,  find  the  num- 
ber of  shares  bought  to  realize  a  profit  of  $225. 

9.  How  many  shares  of  Atchison  bought  at  80-|  and 
sold  at  81f  realize  a  profit  of  $1000? 

10.    How  many  Chicago  B.&Q.  new  4s  at  93^  can  I 
buy  for  $18,650? 


STOCKS  AND  BONDS  251 

Example  l.  By  investing  money  in  4  %  stock  a  man 
realizes  a  profit  of  5  %  on  the  money  invested.  Find 
the  price  of  the  stock. 

SOLUTION.    The  gain  on  $100  invested  is  $5. 

The  gain  on  $20  invested  is  $  1. 

The  gain  on  $80  invested  is  $4. 

The  stock  cost  $80  a  share. 

The  market  price  of  the  stock  is  $80  -  $  4  =  $  79 J. 

A  O  O 

Example  2.  How  much  money  must  be  invested  in 
United  States  old  4s  registered  at  103  to  produce  an 
annual  income  of  $600? 

SOLUTION.  United  States  old  4s  registered  pay  a  divi- 
dend of  $4  a  share. 

The  number  of  4s  in  600  is  150. 

One  hundred  and  fifty  shares  produce  $600. 

The  cost  of  1  share  is  103  +  £. 

The  cost  of  150  shares  is'103£  x  150  =  $15,462.50. 

EXERCISE   136 

1.  If  stock  paying  a  dividend  of  8  %  gives  an  income 
at  the  rate   of   5  %  on  the  money  invested,  what  is  the 
market  value  of  the  stock? 

2.  If  6  %  stock  produces  a  gain  of  4  %  on  the  money 
invested,  find  its  market  value. 

3.  How  much  money  must  I  invest  in  Western  Union 
at  92|,  paying  a  dividend  of  5%,  to   derive   an  annual 
income  of  $1000? 

4.  How  much  must  be   invested   in   Pullman    Palace 
Car  at  218,  paying  a  yearly  dividend  of  8  %,  to  derive  an 
income  of  $1200  a  year  ? 

5.  If  I  buy  4|  %  stock  at  97£,  what  rate  per  cent  do  I 
get  for  my  money  ? 


252  ADVANCED  BOOK  OF  ARITHMETIC 

6.  What  must  I  pay  for  5  %  stock  so  as  to  make  a 
profit  of  8%  on  my  investment? 

7.  How  much  must  be  invested  in  Northwestern,  pay- 
ing 6%,  at  206,  to  derive  an  income  of  $900? 

8.  Which  is  the  more  profitable  investment,  5  %  stock 
at  119 1  or  S%  stock  at  219 J? 

9.  How  much  must  be  paid  for  a  $1000  bond  at  87 J? 

10.  City  bonds  bought  at  89 J  pay  5  %.     What  rate  of 
interest  do  they  pay? 

11.  If  I  have  $  5000  stock  in  United  States  4s  at  102, 
what  annual  dividend  do  I  receive  ?    If  I  sold  my  stock, 
what  should  it  bring? 

12.  How  much  should  be  realized  by  selling  $10,000  of 
each  of   the  stocks  quoted  above  at  the  highest  prices 
paid  March  11,  1903? 

13.  Which   is  the  better  investment,  Illinois  Central, 
paying  6%    dividend,  at  137|,   or    United   States  4s  at 
109f? 

14.  How  many  shares  of  stock  must  a  broker  sell  to 
make  $54.75  brokerage? 

CUSTOMS   AND   DUTIES 

Taxes  levied  on  imported  goods  are  called  duties. 
Duties  are  of  two  kinds,  specific  and  ad  valorem.  The 
latter  is  a  percentage  of  the  foreign  value  of  the  goods; 
the  former  is  a  definite  amount  of  money  on  some  stand- 
ard quantity,  such  as  a  yard,  gallon,  pound,  etc.  Some 
articles  are  subjected  to  both  specific  and  ad  valorem 
duties.  Duties  are  collected  by  United  States  customs 
officers  at  places  known  as  ports  of  entry. 

The  United  States  revenue  from  duties  for  the  year 
ending  June  30,  1906,  amounted  to  $300,251,877.79, 


CUSTOMS  AND  DUTIES  253 

EXERCISE  137 

1.  Find  the  duty  on  3200  Ib.  of  refined  sugar  at  1.95^ 
per  pound. 

2.  Find  the  duty  on  2000  bu.  of  potatoes  at  25^  per 
bushel. 

3.  Find  the  duty  on  1800  Ib.  of  salt  at  12^  per  100  Ib. 

4.  Find  the  duty  on  5  doz.  parasols  at  4s.  Qd.  each  at 
40  %  ad  valorem. 

5.  Find  the  duty  on  12  microscopes  invoiced  at  £8 
each,  the  rate  of  duty  being  45  %  ad  valorem. 

6.  The  duty  on  books  printed  in  English  is  25  %  ad 
valorem.     If  I  import  the  following  books  at  the  prices 
specified,  how  much  duty  in  United  States  money  will  I 
have  to  pay  ? 

Casey's  Elements  of  Euclid     ....  3s.  9d. 

Casey's  Analytical  Geometry  ....  10s.  Od. 

Leatham's  Spherical  Trigonometry  .     .  4s.  Od. 

Lamb's  Infinitesimal  Analysis      .     .     .  12s.  Qd. 

7.  Find  the  duty  on  the  Clarendon  Press  edition  of 
Shakespeare's  plays,  invoiced  at  24s.,  at  25%  ad  valorem. 

8.  A  hardware  merchant  imported  80  doz.  razors  at 
2s.  Qd.  each,  duty  $1.75  per  dozen  and  20%  ad  valorem. 
Find  the  total  duty  paid. 

9.  What  is  the  duty  on  100  yd.    of   treble   ingrain 
carpet  valued  at  90^  a  square  yard,  the  duty  being  22^  a 
square  yard  and  40  %  ad  valorem  ? 

10.  What  is  the  duty  on  40  clocks  valued  at  14.50 
each,  at  40  %  ad  valorem  ?  If  the  clocks  are  retailed  at  f* 
profit  of  30%,  find  the  selling  price  of  these  clocks. 


254  ADVANCED  BOOK  OF  ARITHMETIC 

11.  Find  the  duty  on  120  doz.  linen  cuffs  at  7s.  6d.  a 
dozen,  the  duty  being  40^  a  dozen  and  20%  ad  valorem. 

12.  Find  the  duty  on  a  fur  rug  valued  at  $50,  duty 
35%  ad  valorem.     At  what  price  should  the  rug  be  sold 
so  as  to  make  a  profit  of  20  %  ? 

13.  Calculate   the   duty   levied  on  5  doz.  straw   hats 
valued  at  $30  a  dozen,  the  duty  being  $1  a  dozen  and 
20  %  ad  valorem. 

14.  What  is  the  duty  on  800  yd.  of  silk  valued  at  $1.15 
a  yard  at  60  %  ad  valorem  ?     At  what   price  per  yard 
should  the  silk  be  sold  to  make  a  profit  of  20  %  ? 

15.  Calculate   the  duty  on  12  horses  valued  at  $250 
each  at  25  %  ad  valorem. 

16.  An  Axminster  carpet,  18  ft.  by  12  ft.,  valued  at 
$1.50  per  square  yard,  is  imported.     Find  the  duty  at 
60^  per  square  yard  and  40  %  ad  valorem. 

17.  What  is  the  duty  on  5  doz.  opera  glasses  valued  at 
$3.50  each  at  45%  ad  valorem? 

18.  What  is  the  duty  on  500  Ib.  of  glue  valued  at  50^ 
per  pound,  the  rate  of  duty  being  15^  per  pound  and  20  % 
ad  valorem  ? 

19.  A  suit  of  clothes  imported  from  England  costs  a 
merchant  $38.80.     The  duty  is  60%  ad  valorem.    Taking 
<£!  =  $4.85,  find  the  invoice  price  in  pounds  sterling. 

20.  Find  the  duty  on  1200  yd.  of  linen  invoiced  at  l^d. 
per  yard  at  45  %   ad  valorem.     At  what  price  per  yard 
should  it  be  marked  so  that  the  dealer  may  give  a  dis- 
count of  10  %  and  make  a  profit  of  20  %  ? 


CHAPTER  IV 

INVOLUTION 

IN  the  language  of  mathematics,  3  x  means  three  times 
the  number  x  ;  i.e.  3  x  may  stand  for  three  times  any 
number  whatever. 

a  -f  b  stands  for  the  sum  of  any  two  numbers,  a  and  b. 
a  —  b  stands  for  the  difference  of  two  numbers,  a  and  b.  In 
other  words,  a  —  b  is  the  number  which  when  added  to  b 
gives  a  for  sum. 

ab  stands  for  the  product  of  two   numbers,  a   and   b. 

-  stands  for  the  quotient  obtained  by  dividing  a  by  b.     In 
other  words,  7  is  that  number  which  when  multiplied  by 

b  gives  a  for  product. 

a2  stands  for  the  square  of  a.  a3  stands  for  the  cube  of  a, 
(a  +  ft)2  stands  for  the  square  of  the  sum  of  two  numbers, 
a  and  b. 

(a  —  5)2  stands  for  the  square  of  the  difference  of  two 
numbers,  a  and  b. 

lab  stands  for  seven  times  the  number  ab,  which  is 
itself  the  product  of  two  numbers,  a  and  b. 

— n ! ^—  stands  for  one  half  the  sum  of  two  num- 
bers plus  one  half  their  difference. 

(a  +  b)x  stands  for  the  product  of  a  number,  #,  by  the 
sum  of  two  numbers,  a  and  b. 

255 


256  ADVANCED  BOOK  OF  ARITHMETIC 

(a  —  b*)x  stands  for  the  product  of  a  number,  a?,  by  the 
difference  of  two  numbers,  a  and  b. 

EXERCISE   138 


Add: 

1.    2z 

2.    ly       3. 

4  a       4. 

9b       5.    4  z 

6.    11  a 

3  x 

8y 

9  a 

Ib             Iz 

la 

Bx 

9y 

11  a 

Sb             9z 

5  a 

7.    4  ab 

8.      9  2^ 

9.    4  fo 

10.    12  ab 

11.    9  ab 

Bab 

Ixy 

lie 

IBab 

lab 

7  ab 

8xy 

11  be 

10  ab 

Bab 

12.    T  J<?    13.    aa;    14.    10  x    15.    11  6    16.    12  m    17.    11  a? 

8  be  bx  ax  ab  am  bx 

9  be 

EXERCISE   139 

1.  7a;-4aj=?     7.  15  a- 11  a  =  ?  13.   2  xy  -  xy  =  ? 

2.  6rr-a;=?        8.  17  a  -12  a=?  14.   14  a#  -  10  xy  =  ? 

3.  lla:-8a;=?     9.  18a-13a  =  ?  15.   17a:y-9ajy  =  ? 

4.  12^-2a;  =  ?  10.  12aJ-3a6  =  ?  16.   19xy-7xy  =  ? 

5.  5^-3o;  =  ?     11.  Ha6-4aJ=?  17.  ax-bx=? 

6.  12a  — a  =  ?     12.  5ab-ab  =  ?  18.   cx—dx=l 

19.  ma?  —  nx  —  ? 
Example  1.    Multiply  4  a?  by  5  #. 

SOLUTION.     4#x5#  =  4x#x5xa;  =  4x5xa;X£ 
=  20  z2. 

Example  2.    Multiply  7  x^y  by  3  or?/2. 

SOLUTION.     7  o%  x  3  ^2  =  (7  x  x  x  x  x  y)  x  (3  x  a;  x 


INVOLUTION 


257 


Example  3.  What  is  the  square  of  a  +  ft? 

a  +      5  Multiply    a  -f-  5    by    a,    and   write    the 

a  4-      b  result,    a2  +  aft.       Next   multiply   a  +  ft    by 

a2  +    ab  b  and  write  the  result,  ab  +  ft2.     Then  add. 

+    aft  +  ft2  The  following  is  a  graphical  representation 

a2  +  2aft  +  ft2  of  the  result : 


ab              b 
H 

62 

a2              a 

ab 

A  a  L       b 

Let  AL=  a, 

Then 

Let  ABOD  be  the  square  on  AB,  ALHGr  be  the  square 
on  AL.  Then  HYCKis  the  square  on  ft,  and  G-HKD  = 
LBYH,  because  their  dimensions  are  equal  to  each  other. 
Now,  ABOD  =  ALE  a  +  GHKD  +  HYCK+  LBYH  = 
ALHG-  +  HYOK+  2  G-HKD. 

.*.  (a  +  ft)2  =  a2  +  ft2  +  2  aft.  Hence,  we  have  the  follow- 
ing important  conclusion: 

The  square  of  the  sum  of  two  numbers  equals  the  square 
of  the  first  number  plus  the  square  of  the  second  number  plus 
twice  the  product  of  the  two  numbers. 

The  process  of  finding  the  power  of  a  number  is  called 
involution. 


258  ADVANCED  BOOK  OP  ARITHMETIC 

Example  1.    Square  x  +  9. 

SOLUTION,  (x  +  9)2  =  x2  +  92  +  2  x  9  x  a?  =  a?  +  81  + 
18^  or  a? +18  a; +  81. 

Example  2.     What  is  the  square  of  43  ? 

SOLUTION.  432  =  (40  +  3)2  =  402  +  32  +  2  x  40  x  3  = 
1600  +  9  +  240  =  1849. 

EXERCISE   140 
Find  by  Statement  (3)  the  square  of: 


1. 

10  + 

x. 

7. 

70  + 

b.       13. 

29. 

19. 

72. 

25. 

89. 

2. 

20  + 

x. 

8. 

80  + 

C.         14. 

37. 

20. 

76. 

26. 

92. 

3. 

30  + 

X. 

9. 

90  + 

y.       15. 

47. 

21. 

78. 

27. 

93. 

4. 

40  + 

X. 

10. 

14. 

16. 

54. 

22. 

79. 

28. 

67. 

5. 

50  + 

X. 

11. 

15. 

17. 

62. 

23. 

84. 

29. 

98. 

6. 

60  + 

a. 

12. 

24. 

18. 

68. 

24. 

87. 

30. 

55. 

EXERCISE   141 

1.  Find  the  square  of  .1,  .2,  .3,  .4,  .5,  .6,  .7,  .8,  .9. 

2.  Find  the  square  of  the  reciprocals  of  the  numbers 
from  1  to  20  inclusive. 

3.  Find  the  squares  of  f ,  f ,  f ,  f ,  ^,  11  if,  £. 

4.  Find  the  squares  of  f,  |,  f  11,  If,  If,  21  6|. 

5.  How  does  the  square  of  a  fraction  compare  in,  value 
with  the  fraction  itself? 

6.  Find  the  cubes  of  the   reciprocals  of   the  numbers 
from  1  to  20  inclusive. 

7.  Find  the  value  of  I3  +  23+  33  +  43  +  53  +  63  +  73  +  83 
+  93  + 103. 

8.  Square   each   of   the  following   numbers   and   give 
the  product  correct  to  four  decimal  figures:   (a)   1.732, 
(6)  9.256,  O)  5.401,  (d)  8.129,    (e)  .6834,    (/)  .7609, 
(#)  9.482,  (K)  .7071,  (i)  .7746,  (j)  .9487. 


EVOLUTION  259 


EVOLUTION  —  SQUARE   ROOT 

By  the  square  root  of  a  number  is  meant  that  number 
which  when  squared  produces  the  given  number.  Thus, 
4  is  the  square  root  of  16,  since  42  =  16.  7  is  the  square 
root  of  49  for  a  similar  reason. 

The  square  root  is  defined  also  as  one  of  the  two  equal 
factors  of  a  number.  Thus,  7x7  =  49.  One  of  the 
factors  is  the  square  root  of  49. 

The  symbol  for  square  root,  V>  is  called  the  radical 
sign.  It  is  a  degenerate  form  of  the  first  letter  of  the 
word  radix,  the  Latin  word  for  root.  The  exponent  |-  is 
also  used  as  a  sign  for  the  square  root.  V49,  49*  are  the 
two  ways  of  indicating  the  same  process,  namely,  the 
extraction  of  the  square  root  of  49.  The  number  written 
under  the  radical  sign  is  called  the  radicand. 

Students  should  fix  firmly  in  mind  :  — 
102  =  100  402  =  1600  702  =  4900 

202  =  400  502  =  2500  802  =  6400 

302  =  900  602  =  3600  902  =  8100 

Example  l.    What  is  the  square  root  of  5329? 

SOLUTION.  By  trial  the  square  root  of  5329  is  more 
than  70  and  less  than  80. 


.'.5339  =  4900  +140  x+x* 

Subtract  4900,  and  get, 
140  x  +  x2  =  439. 
.'.  (140 -f- z>  =  429. 

Since  140  is  contained  in  429  3  times,  try  3  as  a  value  of  x. 
(140  +  3)3  =  429. 
.-.  V5329  =  70+  3=73. 


260  ADVANCED  BOOK  ^OF  ARITHMETIC 

Example  2.    V9025  =  ? 
SOLUTION.     V9025  =  90  +  x. 

.-.  8025  =  8100  +  180  x  +  x*. 
.-.  180  x  +  a^  =  925. 
.-.  (180  +  s>  =  925. 

Since  180  is  contained  in  925  5  times,  try  5  for  the  next 
figure  of  the  root. 

(180  +  5)5  =  925. 
.-.  V9025  =  90+5  =  95. 

In  practice,  the  work  is  contracted  as  follows :  Beginning 
at  the  decimal  point,  point  off  the  figures  of  the  number 
in  periods  of  two  figures  each.     By  trial  find 
9    5     the  greatest  digit  whose  square  is  contained 
90.25     in  the  number  denoted  by  the  period  to  the 
81          left.     Write  it  as  the  first  figure  of  the  root, 


185 


9  25     and  write  also  its  square.     Subtract  the  latter 
_9_2t>     from  the  period  to  the  left  and  bring  down 


the  next  period.  Double  the  part  of  the  root 
just  found  for  trial  divisor.  Find  next  the  number  of 
times  the  trial  divisor  is  contained  in  the  number  denoted  by 
the  remainder  and  the  period  brought  down.  Write  this 
result  in  the  quotient  and  in  the  divisor  and  then  multiply. 

EXERCISE   142 

Find  the  square  root  of  : 


1. 

169. 

7. 

2809. 

13. 

7056. 

19. 

7921. 

2. 

441. 

8. 

3844. 

14. 

7569. 

20. 

6084. 

3. 

625. 

9. 

4225. 

15. 

8464. 

21. 

4761. 

4. 

961. 

10. 

5329. 

16. 

9216. 

22. 

3481. 

5. 

1024. 

11. 

5776. 

17. 

9604. 

23. 

2401. 

6. 

1849. 

12. 

6724. 

18. 

9801. 

24. 

3364. 

EVOLUTION 


261 


Example. 

3    5 
12  74  49 

9 


65 


3  74 
3  25 


4949 


hence 


\/127449  =  ? 

SOLUTION.  Divide  the  figures  of  the 
number  into  periods  of  two  as  in  the  pre- 
vious exercises.  Then  proceed  to  extract 
the  square  root  of  the  number  denoted  by 
the  two  periods  to  the  left. 

The  answer  is  obviously  350  +  some 
number.  Let  x  represent  this  number; 


350  +  x  =  V127449. 
.-.(350  +  *)2  =  127449. 
.-.  122500  +  700  x  +  x*  =  127449. 

.-.  700  x+x*  =  4949. 
...  (700  +  x)x  =  4949. 
Since  700  is  contained  in  4900  7  times,  try  7  as  a  value  of 


x. 


(700  +  7)7  =  4949. 


.-.  V127449  =  350  +  7  =  357. 

Notice  the  trial  divisor  is  twice  the  part  of  the  root 
found.  Therefore  in  a  problem  in  square  root  where  the 
radicand  is  an  integer  consisting  of  five  or  six  figures, 
proceed  in  exactly  the  same  way  as  has  been  done  in  a 
problem  consisting  of  four  figures.  Beginning  once  more, 
the  solution  in  its  contracted  form  stands  as  follows  : 

357 
12  74  49 

9  The  trial  divisor  is  always  twice  the  part 

of  the  root  already  found. 


65 


707 


3  74 
325 


4949 
4949 


262  ADVANCED  BOOK  OF  ARITHMETIC 

EXERCISE   143 

Extract  the  square  root  of: 


1. 

100,489. 

7. 

229,441. 

13. 

474,721. 

2. 

110,224. 

8. 

277,729. 

14. 

501,264. 

3. 

120,409. 

9. 

310,249. 

15. 

654,481. 

4. 

171,396. 

10. 

354,025. 

16. 

772,641. 

5. 

190,096. 

11. 

391,876. 

17. 

819,025. 

6. 

199,809. 

12. 

456,976. 

18. 

826,281. 

Memorize  :  — 

(.I)2  =  .01  (.5)2  =.25  (.9)2  =.81 

(.2)2  =.04  (.6)2  =.36  (.ll)2  =.0121 

(.3)2  =.09  (.7)2  =  .49  (.12)2  =.0144 

(.4)2  =  .16  (.S)2  =.64  (.Ol)2  =  .0001 

To  extract  the  square  root  of  a  decimal,  begin  at  the  deci- 
mal point,  and  proceeding  to  the  right,  point  off  the 
figures  in  periods  of  two  ;  next  proceed  as  if  the  number 
were  an  integer.  Thus,  in  taking  the  square  root  of 
.0225,  first  point  off  in  periods  of  two  figures  each.  This 
gives  .02  25.  Next  extract  the  root  of  the  number  de- 
noted by  the  figures  225.  The  result  is  15.  Hence,  the 
required  root  is  .15. 

To  extract  the  square  root  of  a  number  part  integer  and 
part  decimal,  begin  at  the  decimal  point,  and  proceeding  to 
the  left,  point  off  the  integral  part  in  periods  of  two 
figures  each;  next  point  off  the  decimal  part  in  periods  of 
two  figures  each,  beginning  at  the  decimal  point.  If  there 
are  not  enough  figures  in  the  decimal  part  to  make  an 
exact  number  of  periods,  annex  a  cipher  or  as  many 
ciphers  as  are  necessary  to  make  the  required  number  of 
periods. 


EVOLUTION 


263 


Example.    Extract  i\ 

1.     3     0     3     8     4 
1.  70  00  00  00  00 
1. 

ie  square  root  of  1.7. 

Double  1  for  the  first  trial  divi- 
sor.    Double  13  for  the  next  trial 
divisor.    Then  find  the  next  figure 
of  the  root  is  0.     Write  it  in  the 
root    and    in    the    trial    divisor. 
Then    annex   two   more  ciphers, 
and  find  that  the  next  figure  of 
the  root  is  3,  and  so  on. 

23    70 

69 

2603 
26068 
26076 

10000 

7809 

219100 
208544 

4   1055600 
1043056 

EXERCISE   144 
Extract  the  square  root  of : 


1.  .150932. 

2.  .246016. 

3.  .3448. 


4.  .2909. 

5.  .2632. 

6.  .4616. 


7.  .5319. 

8.  .61575. 

9.  .784. 


10.  .083. 

11.  .062. 

12.  .0037. 


To  extract  the  square  root  of  a  fraction  when  its  numer- 
ator and  denominator  are  perfect  squares  is  a  simple 
matter.  Thus,  the  square  root  of  ||  is  f;  the  square  root 
of  IW,  or  |4,  is  |,  or  1J-. 

b4:~  84'  o"  O 

To  get  the  square  root  of  a  fraction,  take  the  square  root 
of  the  numerator,  and  the  square  root  of  the  denominator, 
and  then  write  the  former  result  for  numerator  and  the 
latter  result  for  denominator.  The  fraction  thus  found  is 
the  required  square  root. 

Example  1.    What  is  the  square  root  of  -||  ? 

SOLUTION.    Vl7  =  4.123;  V36  =  6. 


264  ADVANCED  BOOK  OF  ARITHMETIC 

Example  2.   What  is  the  square  root  of  ||  ? 
SOLUTION.    V23  =  4.796; 


This  is  a  roundabout  way  to  take  the  square  root  of  |~|. 
A  shorter  and  better  way  is  to  reduce  the  fraction  to  an 
equivalent  decimal,  and  then  to  extract  the  square  root  of 
this  decimal. 

To  extract  the  square  root  of  a  fraction  whose  denomi- 
nator is  not  a  perfect  square,  reduce  the  fraction  to  an 
equivalent  decimal  and  then  extract  the  square  root  of 
this  decimal. 

EXERCISE  145 

Extract,  to  three  decimal  figures,  the  square  root  of  : 

1.  1.2.         4.    5.2.         7.    3i.          10.    41  13.    f. 

O  4  O 

2.  4.25.       5.    3.3.         8.    9J.          11.    2|.          14.    f. 

3.  1.1.  6.     5|.  9.     If.  12.     TV  15.     TV 

EXERCISE  146 
PROBLEMS  INVOLVING  SQUARE  ROOT 

1.  The  area  of  a  square  field  is  1  A.     Find  the  length 
in  yards  of  one  of  its  sides. 

2.  The  area  of  a  square  field  is  12  A.     Find  the  length 
in  yards  of  one  of  its  sides. 

3.  The  dimensions  of  a  rectangle  are  289  yd.  and  196 
yd.     Find  the  side  of  an  equivalent  square. 

4.  The  dimensions  of  a  rectangle  are  1|  mi.  and  .7  mi. 
Find,  correct  to  four  decimal  figures,  the  side  of  an  equiva- 
lent square. 


EVOLUTION 


265 


5.  Find  in  rods  the  perimeter  of  a  square  field  whose 
area  is  ^  of  a  square  mile. 

6.  The  area  of  a  rectangle  whose  length  is  twice  its 
breadth  is  10  A.     Find  its  dimensions  in  yards. 

HINT.  Draw  a  diagram  ;  divide  it  into  two  equal  parts  by  a  line  paral- 
lel to  its  width.  Notice  what  each  part  is. 

7.  The  area  of  a  rectangle  whose  length  is  three  times 
its  width  is  20  A.     Find  its  dimensions  in  yards. 

8.  A  square  and  a  rectangle  have  the  same  area,  namely 
40  A.     If  the  length  of  the  rectangle  is  twice  its  width, 
find  in  rods  the  difference  between  their  perimeters. 

The  side  of  a  right  triangle  opposite  the  right  angle  is 
called  the  hypothenuse.  The  other  two  sides  are  called  the 
legs  of  the  right  triangle.  One  of  the  legs  is  called  the 
base  of  the  right  triangle,  and  the  other  leg  is  called 
the  altitude  of  the  right  triangle. 

In  a  right  triangle  the  square  on  the  hypothenuse  is  equal 
to  the  sum  of  the  squares  on  the  two  legs.  This  is  the 
famous  Pythagorean  Theorem. 

Designate  the  sides  of 
the  right  triangle  AB  0  by 
the  letters  #,  5,  c.  (a  -f  5)2 
=  a2  +  52  +  2  ab.  But 
(a  -f-  6)2  =  c2  -f-  4  triangles, 
each  having  for  its 
base  and  altitude  a  and 
b.  AMNR  =  c2  +  4  x  J  ab 


-f  52  +  2  ab  =  c2  +  2  ab. 


266  ADVANCED  BOOK  >  OF  ARITHMETIC 

Example.     In  a  right  triangle  the  legs  are  7  and  24. 
Find  the  hypothenuse. 

SOLUTION.  a2  +  b*  =  c2. 


.-.49  +  576  =  <?2. 


.-.  £  =  V625=25. 

EXERCISE   147 

1.  In  a  right  triangle,  given  a  =  6,  5  =  8,  find  c. 

2.  In  a  right  triangle,  given  a  =  5,  5  =  12,  find  c. 

3.  In  a  right  triangle,  given  a  =  8,  b  =  15,  find  <?. 

4.  In  a  right  triangle,  given  a  =  20,  5  =  21,  find  c. 

5.  In  a  right  triangle,  given  a  =•  56,  5  =  90,  find  c. 

6.  In  a  right  triangle,  given  a  =  20,  b  =  99,  find  <?. 

7.  In  a  right  triangle,  given  a  =  17,  b  =  144,  find  c. 

8.  In  a  right  triangle,  given  a  =  39,  6  =  80,  find  c. 

9.  In  a  right  triangle,  given  a—  51,  b  =  140,  find  c. 

10.  In  a  right  triangle,  given  a  =  44,  b  —  52.5,  find  c. 

11.  In  a  right  triangle,  given  a  =  87,  5  =  416,  find  c. 

12.  In  a  right  triangle,  given  a  =  136,  5  =  273,  find  c. 

13.  In  a  right  triangle,  given  a  =  145,  b  =  408,  find  c. 

14.  In  a  right  triangle,  given  a  =  207,  b  =  224,  find  c. 

15.  A  ladder  is  placed  14  ft.  from  a  wall  48  ft.  high. 
How  long  must  the  ladder  be  to  reach  to  the  top  of  the 
wall? 

16.  Find  the  length  of  the  diagonal  of  a  square  if  one 
side  of  the  square  is  10  rods. 

17.  Find  the  length  of  the  diagonals  of  a  rectangle,  the 
dimensions  of  the  rectangle  being  17  rd.  and  25  rd. 


AREAS  OF   PLANE  TRIANGLES  267 

Example.  If  the  hypothenuse  of  a  right  triangle  is 
493  and  one  leg  is  468,  find  the  other  leg. 

SOLUTION.     Let  the  required  leg  be  a. 
Then,  a2  +  4682  =  4932. 

.-.  a2  +  219,024  =  243,049. 

Subtract  219,024  from  each  member  of  this  equation. 
.-.  a2  =24,025. 
/.  a  =  155. 

EXERCISE  148 

1.  Hypothenuse  =  377,  base  =  345,  find  the  altitude. 

2.  Hypothenuse  =  545,  base  =  513,  find  the  altitude. 

3.  Hypothenuse  =  449,  base  =  351,  find  the  altitude. 

4.  Hypothenuse  =  5.05,  base  =4.56,  find  the  altitude. 

5.  Hypothenuse  =  .461,  base  =  .38,  find  the  altitude. 

6.  Hypothenuse  =  .481,  alt.  =  .36,  find  the  base. 

7.  Hypothenuse  =  .641,  alt.  =  .609,  find  the  base. 

8.  Hypothenuse  =  .773,  alt.  =  .195,  find  the  base. 

9.  Hypothenuse  =  .697,  alt.  =  .528,  find  the  base. 

AREAS  OF  PLANE   TRIANGLES 

The  following  rule  gives  the  area  of  any  triangle  : 

(1)  Add  the  three  sides  and  take  half  the  sum. 

(2)  Subtract  each  side  separately  from  the  half  sum. 

(3)  Find  the  continued  product  of   the  three  remain- 
ders and  the  half  sum. 

(4)  The  square  root  of  this  product  is  the  area. 

The  proof  of  this  rule  is  too  difficult  to  be  given  in  an 
elementary  arithmetic.  This  rule  enables  one  to  find  the 
area  of  a  tract  of  land  such  as  a  farm. 


268  ADVANCED   BOOK  OF  ARITHMETIC 

Example.     Find  the  area  of  a  triangle  whose  sides  are 
34  ch.,  65  ch.,  and  93  ch. 

SOLUTION 

34        96         96         96 
65        34        65        93 
_93        62         31  3 

2)192 

"96         Area  =  V96  x  62  x  31  x  3  =  V553535  =  744. 
.*.  area  =  744  sq.  ch.  =  74.4  A. 

EXERCISE   149 
Find  the  area  of  each  of  the  following  triangles  : 

l.  Given  the  sides,  13,  20,  21. 

2. .  Given  the  sides,  13,  30,  37. 

3.  Given  the  sides,  33,  34,  65. 

4.  Given  the  sides,  35,  52,  73. 

5.  Given  the  sides,  29,  60,  85. 

6.  Given  the  sides,  140,  143,  157. 

7.  Given  the  sides,  507,  603,  721. 

8.  Given  the  sides,  46  rd.,  75  rd.,  109  rd. 

9.  Given  the  sides,  40  rd.,  51  rd.,  77  rd. 

10.  Given  the  sides,  3.5  ch.,  10  ch.,  11.7  ch. 

11.  Given  the  sides,  5.6  ch.,  6.1  ch.,  7.5  ch. 

12.  Find  area  of  a  triangle,  each  side  being  10  rd. 

13.  Find  area  of  a  triangle,  each  side  being  50  rd. 

14.  Find  the  area  of  an  isosceles  right  triangle  if  the 
hypothenuse  is  27  inches. 

15.  Find  the  area  of  a  square  whose  diagonal  is  72  feet. 

16.  Find  the  side  of  a  square  equivalent  to  the  difference 
of  two  squares  whose  sides  are  89  feet  and  68  feet. 


1 

MENSURATION  OF  THE  CIRCLE  269 

MENSURATION  OF   THE   CIRCLE,   ETC. 

Take  a  string  and  nnv  +^*  length  of  the  circumference 
of  a  circle.  Take  another  string  and  find  the  length  of  the 
diameter  of  the  circle.  Divide  the  former  result  by  the 
latter  to  get  the  ratio  of  the  circumference  of  the  circle  to 
its  diameter. 

Let  O  =  area  of  circle. 
r  =  radius. 
c  =  circumference. 
d  =  diameter. 

The  ratio  of  the  circumference  of  a  circle  to  its  diameter 
is  approximately  3.14159265.  This  ratio  is  denoted  by  the 
Greek  letter  TT  (Pi).  In  cases  where  the  numbers  involved 
are  not  very  large,  or  where  extreme  accuracy  is  not  de- 
manded, 3^  is  a  sufficiently  accurate  approximation  of  TT. 
The  ratio  355  : 113  is  a  close  approximation  to  the  value 
of  TT.  3.1416  is  generally  taken  as  the  value  of  TT.  In 
this  chapter  we  shall  consider  TT  =  3.1416. 

Since  -  =  TT,  .  •.  c  =  ird  =  TT  (2  r)  =  2  TTT. 
d 

Express  in  words  the  relation  c  —  jrd. 
Express  in  words  the  relation  c  =  2  Trr. 

EXERCISE  150 

Find  the  circumference  when: 

1.  The  diameter  is  22.  6.  The  radius  is  67, 

2.  The  diameter  is  46.  7.  The  radius  is  86. 

3.  The  diameter  is  150.  8.  The  radius  is  3.6. 

4.  The  diameter  is  164.  9.  The  radius  is  5.9. 

5.  The  diameter  is  196.  10.  The  radius  is  7.3. 


270  ADVANCED  BOOK  OF  ARITHMETIC 

11.  Find  d  when  c  =  320.44.    15.  Find  r  when  c  =  377. 

12.  Find  d  when  c  =  477.52.    16.  Find  r  when  0=  53.41. 

13.  Find  c?  when  c  =  24.50. '    V/.   Find  r  when  c  =  60.319. 

14.  Find  d  when  c  =  41.7.       18.  Find  r  when  e  =  42.097. 
19.    The  diameter  of  the  front  wheel  of  a  carriage  is 

3  ft.  6  in.     How  many  times  does  the  wheel  revolve  in 
going  1  mi.  ?     Take  3|  as  the  value  of  IT. 

AREA  OF  A  CIRCLE 

A  sector  of  a  circle  is  a  portion  of  a  circle  bounded  by 
two  radii  and  their  included  arc. 

If  the  arc  of  a  sector  is  very  small,  the  sector  will  not 
sensibly  differ  from  a  triangle.  Hence,  the  area  of  a  sec- 
tor equals  J  the  arc  multiplied  by  the  radius,  and  as  a  cir- 
cle may  be  divided  into  a  large  number  of  sectors,  hence, 

C=  J-  cr,  but  c  =  2  TIT. 
.-.  (7=1  x  27rrxr  =  7rr2. 
Also,  since  r2  =  (l  <T)2  =  |  d2, 

.-.  7rr*  =  ^cP. 
4 

Hence,  the  three  rules  for  finding  the  area  of  a  circle : 

(1)  Multiply  one  half  the  circumference  by  the  radius. 

(2)  Multiply  the  square  of  the  radius  by  TT. 

(3)  Multiply  the  square  of   the  diameter  by  ^  IT. 
There  is  a  fourth  rule  for  finding  the  area  of  a  circle. 

<?=  2?rr.     .•.  r  =  - 


2 


7T 


...  7rr2==7rx  __=        xc2  =  .07958^. 
4?r2      4?r 

That  is,  the  area  of  a  circle  equals  the  square  of  its  circum- 
ference multiplied  by  .07958. 


AREA  OF  A  CIRCLE  271 

EXERCISE   151 

1.  Given  r  =  14,  find  0.       ll.    Given  d  =  74,  find  C. 

2.  Given  r=22,  find  0.        12.    Given  d=92,  find  0. 

3.  Given  r=36,  find  (7.        13.    Given  c  =  100,  find  (7. 

4.  Given  r=  4.7,  find  O.       14.    Given  c?  =  7  8,  find  (7. 

5.  Given  r  =  6.5,  find  (7.      15.    Given  <?=83,  find  (7. 

6.  Given  r=8.6,  find  (7.      16.    Given  c=93,  find  (7. 

7.  Given  r  =  9.7,  find  (7.      17.    Given  <?=8.7,  find  (7. 

8.  Given  d=78,  find  (7.       18.    Given  <?=6.9,  find  (7. 

9.  Given  d=  64,  find  <7.       19.    Given  <?=  9.8,  find  (7. 
10.    Given  d=96,  find-  O.       20.    Given  c  =  10.8,  find  (7. 
HJxample  1.    Given  the  area  of  a  circle  equal  to  535.08, 

find  the  radius  of  the  circle. 
SOLUTION.  ?rr2  =  0. 

.-.  3.1416  r2  =  535.08. 

...  ,«=f|||-no.si. 

.*.  r  =  13.05,  nearly. 

Example  2.    Given  the  area  of  a  circle  equal  to  658.98, 
find  the  circumference  of  the  circle. 
SOLUTION.     .07958  c2=  (7. 

.07958c2=658.98 


.*.  <?=91,  nearly. 

EXERCISE   152 

1.  Given  O.=  3019.1,  find  r. 

2.  Given  (7=907.9,  find  r. 

3.  Given  0=  3421.  2,  find  r. 

4.  Given  0=  5541.8,  find  r. 

5.  Given  (7=21.24,  find  r. 


272 


ADVANCED  BOOK  OF  ARITHMETIC 


6.  Given  (7=40.715,  find  d. 

7.  Given  (7=66.476,  find  d. 

8.  Given  (7=  75. 43,  find  d. 

9.  Given  (7=109.36,  find  d. 

10.  Given  (7=141.03,  find  d. 

11.  Given  (7=4.5964,  find  c. 

12.  Given  (7=6.883,  find  c. 

13.  Given  (7=  779. 94,  find  c. 

14.  Given  (7=57,495,  find  <?. 

15.  Given  (7=  33,621,  find  <?. 

16.    How  long  must  a  rope  be  so  that  by  tying  one  end 
of  it  to  a  stake  driven  into  the  ground  and  fastening  the 
other  end  to  a  cow's  horn,  the  cow 
may  graze  over  1  A.  ? 

17.  ABOD  is  a  square  described 
in  a  circle ;    MNRS  is  a  square  de- 
scribed about  the  circle.    If  the  radius 
of  the  circle  is  12,  find  the  areas  of 
ABOD,  MNRS,  and  of  the  circle. 

18.  If  the  diameter  of  a  circle  is 
34  in.,  find  the  difference  between  the 

area  of  the  circle  and  of  the  square  described  in  the  circle. 

19.  If  the  diameter  of  a  circle  is  88  in.,  find  the  differ- 
ence between  the  area  of  the  square  de- 
scribed about  the  circle  and  the  area  of 

the  circle. 

20.  A  regular  hexagon  is  a  figure  of 
six  sides  each  of  the  same  length,  and 
its  six  angles  are  equal  to  one  another. 
A  hexagon  may  be  broken  up  into  six 
equilateral  triangles. 

If  the  side  of  a  regular  hexagon  is  10,  find  its  area. 


AREA  OF  A  CIRCLE 


273 


21.  If  a  regular  hexagon  is  described  in  a  circle,  its  side 
equals  the  radius  of  the  circle.     If  the  radius  of  a  circle  is 
16  in.,  find  the  difference  between  the  areas  of  the  circle 
and  the  regular  hexagon  described  in  the  circle. 

22.  A  regular  hexagon  and  a  square  have  each  a  perim- 
eter of  60  in.     Find  their  areas. 

23.  The  circumference  of  a  circle  equals  the  perimeter 
of  a  square.     Find  which  has  the  larger  area. 

Take  any  angle,  J.,  formed  by  two  radii  in  a  circle,  and 
take  another  angle,  B,  formed  by  two  radii  of  the  same 
circle,  then  designate 
the  subtended  arcs  by 
arc  A  and  arc  B,  then 
A  :  B  =  arc  A  :  arc  B. 
The  reason  for  this  is : 
If  A  were  twice  B, 

then  arc  A  would  be  FIG.  i.  FIG.  2. 

twice  arc  B.  If  A  were  three  times  B,  then  arc  A  would 
be  three  times  arc  5,  and  so  on  for  any  number  of 
times  (Fig.  1). 

A  _  arc  A 
B     arc  B 
or  A  :  B  =  arc  A :  arc  B.     (a) 

If  two  diameters,  mn,  sr  (Fig.  2),  are  drawn  at  right 
angles,  the  four  angles  at  the  center  are  all  right  angles. 
If  each  of  the  right  angles  at  0  is  divided  into  90  equal 
parts,  each  part  is  equal  to  1°.  The  arcs  these  angles 
intercept  on  the  circumference  are  all  equal,  and  as  the 
360th  part  of  the  circumference  is  called  a  degree  on  the 
circumference,  hence  (a)  the  number  of  degrees  in  an 
angle  at  the  center  is  equal  to  the  number  of  degrees  in 


274  ADVANCED  BOOK  OF  ARITHMETIC 

its  arc  on  the  circumference.  This  fact  is  generally  stated 
as  follows :  A  central  angle  is  measured  by  its  intercepted 
arc. 

Example.  The  radius  of  a  circle  is 
15  in.  Find  the  length  of  an  arc  of  37° 
30'  of  the  circumference  of  this  circle. 

SOLUTION.  360°  :  37°  30'  =  length 
of  the  circumference  :  length  of  the 
arc  mn. 

.-.  c=2x  3.1416  x  15  in.  =  94.248  in. 
.-.  360  :  37 J  =  94.248  in.  :  arc  mn. 
.-.  360  x  arc mn=  94.248  x  37J  in. 


94.248  x  37J  in.         Q0  . 

.*.  arc  mn  =  —  -  =9.82  in.,  nearly. 

360 

EXERCISE    153 

1.  The  radius  of  a  circle  is  37  in.     Find  the  length  of 
an  arc  of  72°  of  this  circle. 

2.  The  radius  of  a  circle  is  94  in.     Find  the  length  of 
an  arc  of  30°  of  this  circle. 

3.  What  is  the  length  of  an  arc  of  1°  on  a  circle  whose 
radius  is  58  ft.  ? 

4.  What  angle  does  an  arc  of  40.212  ft.  subtend  at  the 
center  of  a  circle  whose  radius  is  64  ft.  ? 

5.  What  angle  does  an  arc  of  6.032  ft.  subtend  at  the 
center  of  a  circle  whose  radius  is  96  ft.? 

6.  The  distance  of  the  moon  from  the  earth  is  239,000 
mi.,  and  the  diameter  of  the  moon  is  2170  mi.     To  an 
observer  on  the  earth,  what  angle  does  the  moon's  diame- 
ter subtend  ? 


SIMILAR  FIGURES 


275 


SIMILAR  FIGURES 
Similar  figures  are  figures  having  the  same  form. 


Examples.  The  triangles  ABO,  LMN,  are  similar 
triangles. 

All  regular  polygons  of  the  same  number  of  sides  are 
similar  figures.  The  drawing  which  a  surveyor  makes  of 
a  tract  of  land  is  similar  to  the  tract  of  land. 

It  is  shown  in  geometry  that  corresponding  dimensions 
of  similar  figures  have  the  same  ratio;  also  that  the  areas 
of  similar  figures  are  to  each  other  as  the  squares  of  their 
corresponding  dimensions. 

Example  l.  When  a  pole  6 
ft.  high  casts  a  shadow  5  ft., 
how  high  is  a  steeple  whose 
shadow  is  90  ft.? 

SOLUTION.    Let   AB    repre- 
sent  the   pole,   AO 
its    shadow,    x    the    , 
steeple,  and  MR  its 
shadow.     Then 
5  :  90  =  6  :  x.  6 

.-.    3=108.        Ana. 

108  ft. 

A 


276  ADVANCED  BOOK  OF  ARITHMETIC 

Example  2.  The  area  of  a  triangle,  one  of  whose  sides 
is  5  rd.,  is  11  sq.  rd.  Find  the  corresponding  side  of  a 
similar  triangle  whose  area  is  three  times  as  great. 

SOLUTION.     Let  X  equal  the  required  side. 

The  area  of  the  first  triangle  :  the  area  of  the  second 
triangle  =  square  of  the  side  of  the  first  triangle  :  square 
of  the  side  of  the  second  triangle. 


.e. 


JT  =  V75  =  8.662.     Ans.  8.662  rd. 

EXERCISE   154 

1.  When  a  tree  90  ft.  high  casts  a  shadow  75  ft.  long, 
find  the  length  of  the  shadow  cast  by  a  pole  24  ft.  high. 

2.  How  high  is  an  object  which  casts  a  shadow  110  ft. 
when  a  pole  8  ft.  high  casts  a  shadow  5  ft.  ? 

3.  A  map  is  drawn  to  a  scale  of  40  mi.  to  1  in.     On 
this  map  two  cities  are  2J  in.  apart.     How  many  miles 
are  there  between  these  cities? 

4.  In  a  map  of  a  city  two  public  buildings  are  9J  in. 
distant.     If  the  map  is  drawn  to  the  scale  of  1  in.  to  f  of  a 
mile,  how  far  is  it  from  one  of  these  buildings  to  the  other  ? 

5.  The  area  of  a  triangle  is  15  sq.  ft.,  and  one  of  its 
sides  is  10  ft.     Find  the  corresponding  side  of  a  similar 
triangle  five  times  as  large. 

6.  The  altitude  of  a  triangle  is  10  ft.     If  the  triangle 
is  divided  into  two  equal  parts  by  a  line  parallel  to  its 
base,  how  far  from  the  vertex  must  this  line  be  drawn  ? 


SURFACES  OF  SOLIDS 


277 


7.  Corresponding  sides  of  two  similar  quadrilaterals  are 
in  the  ratio  of  4  to  11.     Find  the  ratio  of  their  areas. 

8.  The   diameters   of  two   circles   are    12   and  18  in. 
Find  the  ratio  of  their  areas. 

9.  The  distance  between  two  cities  is  90  mi.,  and  on  a 
map  containing  both  cities  their  positions  are  distant  5f 
in.     What  area  is  represented  by  a  circle  of  J  in.  radius 
on  this  map  ? 

SURFACES  OF   THE   PRISM,   PYRAMID,   CYLINDER, 
CONE,   AND   SPHERE 

A  right  prism  is  a  solid,  two  of  whose  faces  are  equal  and 
parallel  polygons,  and  whose  other  faces  are  rectangles. 

The  upper  and  lower  faces  are  called  the  bases,  and  the 
other  faces  are  called  lateral  faces. 


PRISMS 


PYRAMIDS 


CONE 


A  right  pyramid  is  a  solid  whose  base  is  a  regular  poly- 
gon, and  whose  other  faces  are  triangles  equal  in  area. 

A  cone  is  a  solid  made  by  the  revolution  of  a  right 
triangle  about  one  of  its  legs. 

A  cylinder  is  a  solid  made  by  the  revolution  of  a 
rectangle  about  one  of  its  sides  when  that  side  is  fixed. 


278  ADVANCED  BOOK  OF  ARITHMETIC 

The  following  rules  may  be  easily  established  experimen- 
tally by  paper  cutting  or  other  devices  : 

The  lateral  surface  of  a  right  prism  equals  the  product  of 
the  perimeter  of  its  base  by  the  height  of  the  prism. 

The  lateral  surface  of  a  right  pyramid  equals  one  half  the 
perimeter  of  its  base  by  the  altitude  of  one  of  its  lateral 
faces. 

The  convex  surface  of  a  cone  equals  one  half  the  circum- 
ference of  its  base  by  its  slant  height. 

The  convex  surface  of  a  cylinder  equals  the  circumference 
of  its  base  by  its  height. 

The  surface  of  a  sphere  equals  four  times  the  area  of  one 
of  its  great  circles,  i.e.  4  irr2. 

EXERCISE  155 

1.  Find  the  lateral  surface  of  a  quadrangular  prism,  the 
dimensions  of  whose  base  are  16  ft.  by  8  ft.,  and  whose 
height  is  12  ft. 

2.  Find  the  area  of  the  walls  of  a  room,  having  given 
the  dimensions  of  the  floor  as  18  ft.   by  16  ft.,  and  the 
height  as  10  ft. 

3.  Find  the  height  of  a  triangular  prism,  the  sides  of 
its  base  being  5,  6,  and  7  ft.,  and  its  lateral  area  being  190 
sq.  ft. 

4.  Find  the  height  of  a  pentagonal  prism  whose  lateral 
area  is  300  sq.  ft.,  and  each  side  of  whose  base  is  8  ft. 

*  5.  The  base  of  a  square  pyramid  is  40  ft.  long,  and 
the  altitude  of  each  of  its  triangular  faces  is  26  ft.  Find 
its  lateral  area.  Find  the  cost  of  painting  its  lateral 
surface  at  2J^  per  square  foot. 


SURFACES  OF  SOLIDS  279 

6.  Each  side  of  a  hexagonal  pyramid  is  14  ft.  and  its 
slant   height   is   15   ft.      Find   the   area    of    its    lateral 
surface. 

7.  A  pyramidal  tent  whose  base  is  a  square  22  ft.  on  a 
side  has  a  slant  height  of  30  ft.      Find  the  cost  of  the 
canvas  for  the  tent,  at  18^  per  square  yard. 

8.  Find  the  number  of  square  yards  in  the  lateral 
surface  of  a  triangular  pyramid,  each  side  of  the  base 
being  21  ft.  and  the  slant  height  being  42  ft. 

9.  The  radius  of  the  base  of  a  right  cone  is  49  in. 
and  the  slant  height  is  50  in.     Find  its  convex  surface 
in 


the 
hei 


ne, 
int 


3  3  a 

CAife.  ^  -^  x«.  ana.  the 

slant  height  16  ft.  ? 

12.  Find  the  convex  surface  of  a  cylinder  the  diameter 
of  whose  base  is  19  ft.  and  whose  height  is  50  ft. 

13.  Find  the  convex  surface  of  a  cylinder,  the  radius  of 
the  base  being  41  in.,  and  the  height,  60  in. 

14.  A  standpipe  has  a  diameter  of  30  ft.  and  is  150  ft. 
high.    Find  the  cost  of  painting  it  at  25  ^  per  square  yard. 

15.  Find  the  surface  of  a  sphere  whose  radius  is  98  in. 

16.  Find  the  surface  of  a  sphere  if  its  diameter  is  42  in. 

17.  The  diameter  of  the  planet  Mercury  is  3030  mi.; 
find  the  area  of  the  planet. 

18.  The  diameter  of  the  planet  Venus  is  7700  mi. ;  find 
the  area  of  the  planet. 


280  ADVANCED  BOOK  OF  ARITHMETIC 

19.  The  diameters  of  the  major  planets  are,  respectively, 
86,000,  73,000,  32,000,  33,000  mi.     Find  the  number  of 
million  square  miles  in  the  area  of  each  of  these  planets. 

20.  The  surface  of  a  sphere  is  10,568  sq.  in.     Calculate 
its  diameter. 

VOLUMES  OF   SOLIDS 

The  volume  of  a  rectangular  prism  is  equal  to  the  area  of 
its  base  multiplied  by  its  height. 

The  volume  of  a  cylinder  is  also  equal  to  the  product  of 
its  base  by  its  height. 


The  volume  of  a  pyramid  and  that  of  a  cone  are  each 
equal  to  one  third  the  product  of  the  area  of  the  base  and 
height. 

The  volume  of  a  cylinder  having  r  for  radius  of  base 
and  2  r  for  height  is  Tir2  x  2  r  —  2  Trr3. 

The  volume  of  a  sphere  having  r  for  radius  is  f  of  2  Trr2 
=  |-  Trr3,  and  since  r  —  \  d,  .'.  rs  =  -|  c?3 ; 

...A7rr3  =  A7rX|  J3  =  |  C?3=. 5236^. 

EXERCISE  156 

1.  Find  the  volume  of  a  triangular  prism,  the  sides  of 
the  base  being  11,  25,  30  in.,  respectively,  and  the  height 
of  the  prism  being  40  in. 


VOLUMES  OF  SOLIDS  281 

2.  Find  the  volume  of  a  square  pyramid,  if  the  sides 
of  the  base  are  each  equal  to  10  in.,  and  the  height  is  21  in. 

3.  Find  the  volume  of  a  cone,  the  radius  of  its  base 
being  12  in.,  and  its  height  being  27  in. 

4.  Find  the  volume  of  a  cone,  if  the  radius  of  the  base 
is  25  in.,  and  the  height  is  24  in. 

5.  Find  the  volume  of  a  hexagonal  pyramid,  each  side 
of  its  base  being  10  in.,  and  its  height  being  30  in. 

6.  Find  the  volume  of  a  sphere  whose  radius  is  20  in. 

7.  Find  the  volume  of  a  sphere,  the  radius  being  8  ft. 

8.  How  many  gallons  does  a  cylindrical  cistern  hold, 
the  diameter  of  its  base  being  9  ft.  4  in.,  and  its  height 
8ft.? 

9.  How  many  gallons  does  a  cylindrical  cistern  contain, 
if  the  diameter  of  its  base  is  11  ft.  and  its  height  is  6  ft. 
5  in.? 

10.  The  diameter  of  the  base  of  a  cylinder  is  10  in., 
and  its  height  is  10  in.     Find  the  ratio  of  the  volume  of 
this  cylinder  to  the  volume  of  a  sphere  10  in.  in  diameter. 

11.  The  diameter  of  the  base  of  a  cone  is  1  ft.  and  its 
height  is  1  ft.     Find  the  ratio  of  the  volume  of  this  cone 
to  the  volume  of  a  sphere  whose  diameter  is  1  ft. 

12.  The  surface  of  a  cube  contains  84  sq.  ft.  54  sq.  in. 
Find  its  volume. 

13.  The  base  of  a  pyramid  is  a  triangle  whose  sides  are 
1  ft.  1  in.,  3  ft.  1  in.,  3  ft.  4  in.,  and  whose  volume  is  1  cu. 
ft.  1152  cu.  in.     Find  its  height. 

14.  The  surface  of  a  sphere  equals  1257  sq.  in.     Find 
its  volume. 

15.  The  surface  of  a  hemispherical  dome  is  2513.5  sq. 
ft.     Find  its  diameter. 


282 


ADVANCED  BOOK  OF  ARITHMETIC 


16.  The  volumes  of  similar  solids  are  to  each  other  as 
the  cubes  of  their  corresponding  dimensions.     How  many 
times  as  large  as  the  earth  is  the  sun  ?     The  diameter  of 
the  sun  is  nearly  888,000  mi.,  and  the  diameter  of  the 
earth  is  nearly  8000  mi. 

17.  Find  how  many  times  as  large  as  the  moon  is  the 
earth.     The  moon's  diameter  is  2200  mi.,  nearly. 

18.  How  many  times  as  large  as  the  earth  is  Saturn? 
The  diameter  of  Saturn  is  73,000  mi. 

19.  How  many  times  as  large  as  the  earth  is  Jupiter? 
The  diameter  of  Jupiter  is  88,000  mi.,  nearly. 


120- 


70 


50 

40 

30 
20 


30 


MEASURE   OF   TEMPERATURE 

A  thermometer  is  an  instrument  for  meas- 
uring heat.  The  principle  of  the  thermom- 
eter is  that  substances  expand  with  heat, 
according  to  a  natural  law. 

There  are  two  different  styles  of  thermom- 
eter in  general  use,  —  the  Centigrade  and  the 
Fahrenheit.  The  Centigrade  thermometer 
marks  the  melting  point  of  ice  0°,  and  the 
boiling  point  of  water  100°.  The  interval  be- 
tween these  points  is  divided  into  100  parts, 
or  degrees,  so  that  the  change  in  the  volume 
of  the  mercury  between  any  two  consecutive 
marks  is  T-^  of  the  change  from  0°  to  100°. 

The  Fahrenheit  thermometer  divides  the 
interval  from  the  melting  point  of  ice  to  the 
boiling  point  of  water  into  180°.  It  marks 
the  melting  point  of  ice  32°,  and  the  boiling 
point  of  water  212°. 


MEASURE  OF  TEMPERATURE  283 

Notation.     92°  C.  means  92  degrees  on  the  Centigrade 
thermometer. 

45°  Fahr.  means  45  degrees  on  Fahrenheit's  thermometer. 
-f- 10°  means  10  degrees  above  zero. 
—  10°  means  10  degrees  below  zero. 
Verify  by  counting  32°  backward. 

20°  -  32°  =  -  12°. 
10°  -  32°  =  -  22°. 
-  2°  -  32°  =  -  34°,  etc. 

(1)  To  change  from  degrees   Fahrenheit   to   degrees  Centi- 
grade, subtract  32°  and  multiply  the  remainder  by  -|. 

(2)  To   change  from  degrees  Centigrade  to  degrees  Fahren- 
heit, multiply  the  number  of  degrees  Centigrade  by  |  and  add 
32  to  the  product. 

Explanation  of  the  rules: 

(1)  Suppose  the  temperature   on   a   Fahrenheit   ther- 
mometer is  n  degrees.     Subtract  32°  to  get  the  number  of 
degrees  from  0.     A  difference  of  180°  Fahrenheit  =  a  dif- 
ference of  100°  Centigrade.     Therefore,  a  difference  of  1° 
Fahrenheit  =  a  difference  of  |°  Centigrade.     Therefore,  a 
difference   of    (n  -  32°)   Fahrenheit  =  f  (n  -  32°)    Centi- 
grade, which  symbolizes  the  first  of  the  above  rules. 

(2)  A  difference  of  n°  C.  =  a  difference  of  |  n°  Fahren- 
heit.    Hence,  n°  C.  =  (f  n°  +  32°)  Fahr. 

EXERCISE    157 

Express  the  following  Fahrenheit  temperatures  on  the 
C.  scale  : 


1.    86°. 

4.    248°. 

7.    38°. 

10.    -13°. 

2.    77°. 

5.    68°. 

8.    23°. 

11.    -40°. 

3.    203°. 

6.    54°. 

9.    15°. 

12.    -90°. 

284  ADVANCED  BOOK  OF  ARITHMETIC 

Express  the  following  C.  temperatures  on  the  Fahr. 
scale  : 

13.  35°.  16.    20°.          19.     -10°.       22.    -18°. 

14.  55°.  17.    18°.          20.    -20°.       23.    -24°. 
is.   25°.             is.    8°.  21.     -14°.      24.    -273°. 

TABLE  OF  MELTING  POINTS 

Mercury  .-  40°  C.    Lead    .    .  326°  C.     Gold    ....    1035°  C. 
Sulphur.    113°  C.-   Zinc     .    .  415°C.     Cast  iron,  1100°  to  1200°  C. 

25.  Give  the  above  table  in  the  Fahrenheit  scale. 

26.  Water  attains  its  maximum  density  of  4°  C.     Ex- 
press this  temperature  on  Fahrenheit's  scale. 

THE  METRIC   SYSTEM  OF  WEIGHTS   AND  MEASURES 

The  metric  system  is  now  used  by  more  than  forty 
countries,  and  it  is  the  only  system  used  in  text-books  of 
science.  Upward  of  twenty  nations  contribute  to  the  sup- 
port of  the  International  Bureau  of  Weights  and  Measures 
in  Paris.  For  these  reasons  the  Metric  System  deserves 
to  be  called  the  International  System. 

The  only  great  nations  which  have  not  adopted  the 
Metric  System  are  the  United  States  and  Great  Britain. 
In  the  United  States  the  system  is  legalized,  and  none 
other  is  used  in  the  Philippines  and  Porto  Rico. 

The  Metric  System  is  so  called  because  the  meter  is  the 
basis  of  the  system.  The  meter  is  the  standard  unit  of 
linear  measure.  Its  length  is  the  ten-millionth  part  of 
the  distance  from  the  equator  to  the  north  pole  measured 
on  the  meridian  of  Paris.  Its  length  in  this  country  is 
39.37  inches.  In  the  United  Kingdom  the  legal  equivalent 
of  the  meter  is  39.370113  inches  and  on  the  continent  of 
Europe  39.370432  inches. 


METRIC  SYSTEM  OF  WEIGHTS  AND  MEASURES        285 

The  standard  meter  from  which  all  others  are  derived 
is  a  bar  made  of  an  alloy  of  platinum  and  iridium,  kept  in 
the  International  Bureau  of  Weights  and  Measures  in 
Paris.  Duplicates  of  this  standard  meter  have  been  fur- 
nished to  all  the  nations  of  the  world. 

The  Metric  System  is  a  decimal  sj^stem.  In  it  there 
are  no  compound  rules.  It  is  the  simplest  and  the  most 
perfect  system  ever  devised. 

The  names  for  the  multiples  of  the  standard  unit  in  the 
Metric  System  are  formed  from  the  names  of  the  standard 
unit  by  means  of  prefixes  derived  from  the  Greek  words 
meaning  ten,  one  hundred,  one  thousand,  and  ten  thou- 
sand, i.e.  deka,  hekaton,  chilioi,  murioi.  The  names  for  the 
submultiples  of  the  standard  units  are  formed  in  a  similar 
manner  from  the  Latin  words  meaning  ten,  one  hundred, 
one  thousand,  i.e.  decent,  centum,  mille.  Thus: 

Dekameter  means  ten  meters. 
Hektometer  means  one  hundred  meters. 
Kilometer  means  one  thousand  meters. 
Myriameter  means  ten  thousand  meters. 
Decimeter  means  one  tenth  of  a  meter. 
Centimeter  means  one  hundredth  of  a  meter. 
Millimeter  means  one  thousandth  of  a  meter. 

The  standard  units  are  the  meter,  the  liter,  and  the  gram. 
The  gram  being  a  very  small  weight,  the  kilogram  is  most 
used  in  ordinary  affairs.  In  the  Metric  System  the  meter, 
the  liter,  and  the  kilogram  serve  for  everyday  trade  in 
exactly  the  same  manner  as  the  yard,  the  quart  measure, 
and  the  pound  Avoirdupois  in  our  system  of  weights  and 
measures.  The  international  meter  and  the  kilogram  are, 
since  1893,  the  fundamental  standards  of  length  and  mass 
in  the  United  States. 


286 


ADVANCED   BOOK  OF  ARITHMETIC 


The  gram  is  the  weight  of  distilled  water  at  4°  Centi- 
grade, which  fills  a  cubical  vessel,  one  of  whose  edges  is 
one  centimeter. 

inn  iiiiiMii  iiiiiuii  miiiiii  iimim  iiiiiiiii  iiiniiii  iiiunii  iiiiiiin  MINIMI 


01         234         50         7         89        10  cm. 

imliiiilmiimili 


I     I     I     I     I     I     Ml 


0 


4  in. 


COMPARISON  SCALE:  10  CENTIMETERS  AND  4  INCHES.    (ACTUAL  SIZE.) 

LINEAR  MEASURE 
10  millimeters  (mm.)  =  1  centimeter 
10  centimeters  (cm.)    —  1  decimeter 
10  decimeters  =  1  meter 

10  meters  (m.)  =  1  dekameter 

10  dekameters  =  1  hektometer 

10  hektometers  =  1  kilometer 

10  kilometers  (km.)     =  1  myriameter 

The  units  in  common  use  are  the  centimeter,  meter,  and 

kilometer. 

SURFACE  MEASURE 

100  square  millimeters  (qmm.)  =  1  square  centimeter 
100  square  centimeters  (qcm.)     =  1  square  decimeter 


=  1  square  meter 

=  lar 

=  1  hektar 

=  1  square  kilometer  (qkm.) 

An  ar  is  the  area  of  a  square  whose  side  is  ten  meters. 
1  qm.  is  therefore  equal  to  a  centar  (ca.).  The  areas  of 
small  tracts  of  land,  such  as  gardens,  are  expressed  in  ars. 


100  square  decimeters 
100  square  meters  (qm.) 
100  ars 
100  hektars  (ha.) 


METRIC  SYSTEM  OF  WEIGHTS  AND   MEASURES        287 

The  areas  of  larger  tracts,  such  as  farms,  are  expressed  in 
hektars.  The  areas  of  still  larger  tracts,  such  as  countries, 
are  expressed  in  square  kilometers. 

CUBIC   MEASURE 

1000  cubic  millimeters  (cmm.)  =  1  cubic  centimeter 

1000  cubic  centimeters  (ccm.)     =  1  cubic  decimeter,  or  liter 

1000  liters  (1.)  =  1  cubic  meter  (cbm.) 

The  liter  corresponds  to  our  quart.  The  hektoliter 
(hi.)  corresponds  to  our  bushel.  The  cubic  meter  is  also 
called  a  stere. 

WEIGHT 

1000  milligrams  (mg.)   =  1  gram 

1000  grams  (g.)  =1  kilogram,  or  kilo 

1000  kilograms  (kg.)      =  1  tonneau  (T.),  or  ton 

A  gram  represents  the  weight  of  1  ccm.  of  distilled 
water  at  4°  C. 

A  kilogram  is  the  weight  of  1  cu.  dm.  of  distilled  water 
at  4°  C. 

A  tonneau,  or  ton,  is  the  weight  of  1  cbm.  of  distilled 
water  at  4°  C. 

A  quintal  is  100  kg. 

REDUCTION 

^Example  l.  Reduce  75.623  m.  to  centimeters,  and  also 
to  millimeters. 

SOLUTION,  (a)  Since  there  are  100  cm.  in  1m.,  reduce 
meters  to  centimeters  by  multiplying  the  number  of  meters 
by  100.  This  is  done  by  moving  the  decimal  point  two 
places  to  the  right. 

(6)  Since  there  are  1000  mm.  in  1  m.,  therefore  multiply 
the  number  of  meters  by  1000.  This  is  done  by  moving 
the  decimal  point  three  places  to  the  right. 


288  ADVANCED  BOOK  OF  ARITHMETIC 

Ans.     (a)  7562.3  cm. 
(5)  75623mm. 

Example  2.    Reduce  85679  mm.  to  meters. 
SOLUTION.     This  is  done  by  dividing  by  1000,  i.e.  by 
moving  the  decimal  point  three  places  to  the  left. 

Ans.  85.679  m. 
EXERCISE   158 
Add: 

(1)  (2)  (3) 

96458.23  in.  92435.9  cm.  924357.9  mm. 

23458.98  m.  23456.8  cm.  8765.1  mm. 

14329.09  m.  98239.2  cm.  986201.8  mm. 

920184.60  m.  98209823.9  cm.  87623.7  mm. 

1908.02m.  1980098.76cm.  987.9mm. 

90327.92  m.  6543097.1  cm.  23456.2  mm. 

1872095.72m.  7609.9cm.  10098.3mm. 

902376.253  m.  980098.3  cm.  90098.9  mm. 

999999.9  mm. 

4.  A  man  walks  on  four  successive  days.     The  first  day 
he  walks  11.7  km.  ;  the  second  day,  984  m.  ;  the  third 
day,  2950  m.  ;  the  fourth  day,  12.8  km.     How  far  does  he 
travel?     Give  the  answer  in  meters. 

5.  From  25.724  km.  take  6270  m. 

6.  Multiply  11.732  m.  by  12. 

7.  How  many  times  is  25  mm.  contained  in  24  m.  ? 

8.  How  many  times  is  7.03  m.  contained  in   .0494209 
km.? 

9.  How  many  times  is  12  ca.  contained  in  6  ha.  ? 

10.    How  many  farms  of  8  ha.  each  can  be  made  out  of 
a  square  whose  side  is  3  km.  ? 


METRIC  SYSTEM   OF  WEIGHTS  AND  MEASURES       289 

11.  A  vessel  contains  250  ccm.     How  many  such  vessels 
would  hold  5  cbm.? 

12.  How  many  times  is  800  ccm.  contained  in  50  1.? 

13.  How  many  times  is  450  ccm.  contained  in  22.5  hi.? 

14.  A  dime  weighs  2.5g.     How  many  dimes  can  be 
coined  from  3  kg.  standard  silver  ?  how  many  25  ^  pieces, 
weights  being  in  proportion  to  values  ? 

15.  How  often  is  1521  mg.  contained  in  5.9319  kg.  ? 
Example.     Find  the  area  of  a  rectangle,  if  its  length  is 

425.8  m.  and  its  breadth  is  .3256  km. 

SOLUTION.     First,  reduce  .3256  km.  to  meters. 
.3256km.  =  325.6  in. 

Second,  multiply  in  the  usual  manner  425.8 

and   get   for    product   138,640.48    qm.  325.6 

Reduce  this  to  ars  by  dividing  by  100,  25548 

and  reduce  the  ars  to  hektars  by  divid-  21290 

ing   by  100.     Both   operations  can  be  8516 

performed    at    once    by     moving    the  12774 

decimal  point  four  places  to  the  left.  138640.48  qm. 
Am.  13.864048  ha. 

EXERCISE  159 

Find  the  area  of  each  of  the  following  rectangles : 
LENGTH  WIDTH  LENGTH  WIDTH 

1.  625  m.         125  m.          4.    369.4  m.         184.7  m. 

2.  305  m.  61  m.          5.    488.9  m.         244.45  m. 

3.  338  m.         169  m.          6.    5767  m.  11.534  m. 

7.  Find  the  volume  of  a  rectangular  solid  whose  di- 
mensions are  1.2  m.,  .9  m.,  and  .75  m. 

8.  Find  the  area  of  a  triangle  whose  sides  are  14  m., 
48  m.,  and  50  m. 

9.  Find  the  area  of  a  circle  whose  radius  is  .78  m. 

10.    Find  the  volume  of  a  sphere  whose  radius  is  18  cm. 


290  ADVANCED  BOOK  OF  ARITHMETIC 

11.  The  legs  of  a  right  triangle  are  56  cm.  and  105  cm. 
Find  the  hypothenuse. 

12.  Find  the  volume  of  a  rectangular  prism,  the  dimen- 
sions of  the  base  being  14  m.  by  12  m.,  and  height  9.5  in. 

13.  A  boiler  has  300  tubes  2.4  m.  long,  7.5  cm.  diameter. 
What  is  the  area  of  the  tube  heating  surface  ? 

14.  Find  the  weight  of  a  spherical  cast  iron  shell  32.5 
cm.  outside  and  27.5  cm.  inside  diameter. 

15.  An  iron  plate  3  mm.  thick  weighs  1  km.     What  is 
its  area  ? 

The  following  approximations  should  be  fixed  in  mind : 

1  meter  =  39  inches ;  1  kilometer  =  |-  of  1  mile. 

1  centimeter  =  |  of  an  inch ;  1  hektar  =  2|  acres. 

1  liter  =  1  quart ;  1  kilogram  =  2^  pounds  Avoirdupois. 

EQUIVALENTS  OF   COMMON  UNITS  IN    METRIC    UNITS 

1  inch       =  25.4001  mm.  1  sq.  foot       =  .0929  qm. 

1  foot        =  .304801  m.  1  sq.  yard      =  .8361  qm. 

1  yard       =  .914402  m.  1  A.  .4047  ha. 

1  mile       -  1.60935  km.  1  sq.  mi.        =  2.59  qkm. 

1  quart     =  .94636  1.  1  cu.  inch     =  16.3872  ccm. 

1  gallon    =  3.78543  1.  1  cu.  foot      =  .02832  cbm. 

1  bushel   =  .35239  hi.  1  cu.  yard     =  .7646  cbm. 

1  sq.  inch  =  .6452  qcm.  1  Ib.  Avoir.  =  .45359  kg. 

EQUIVALENTS  OF  METRIC   UNITS    IN    COMMON    UNITS 

1  meter  =39.37  in.  1  sq.  centimeter    =  .155  sq.  in. 

1  kilometer  =  .62137  mi.         1  sq.  meter  =  1.196  sq.  yd. 

1  hektar  =2.471  A. 

1  qkm.  =  .3861  sq.  mi. 

1  cu.  centimeter  =  .061  cu.  in. 

1  cu.  meter  =  35.314  cu.  ft. 

1  liter  =  1.0567  qt.*        1  kilogram  =  2.20462  Ib. 

1  hektoliter  =  2.83774  bu.       1  tonneau,  or  ton  =  2204.6  Ib. 

*  Liquid  quarts,  or  0.9081  dry  quarts. 


ANNUAL  INTEREST  291 

ANNUAL  INTEREST 


Lincoln,  Neb., 


dewuvncL  c/  promise  to  pay  to  __________________ 

f&tt  ______________  or  order 


Eight  hundred..  ...°°  Dollars. 

100 


Value  received,  with,  i^nt&^e^t  awtt/uaJUA*  at 
No.  <)2. 


In  some  states  if  a  note  contains  the  words  "with 
interest  annually,"  and  if  the  interest  remains  unpaid  for 
a  number  of  years,  then  the  interest  due  at  the  end  of 
each  year  bears  interest  until  the  date  of  settlement. 

The  interest  by  this  method  of  reckoning  is  called 
annual  interest.  The  interest  we  have  up  to  this  time 
considered  is  known  as  simple  interest. 

Example.  How  much  is  due  on  a  note  for  $  800,  dated 
Jan.  5,  1898,  and  bearing  interest  annually  at  8%,  if  left 
unpaid,  both  in  principal  and  interest,  until  March  10, 
1903? 

SOLUTION 

YB.         MO.      DA. 

Interest  on  $800  for  1  yr.  at  8%  =  $64.       1903     3     10 

Interest  on  $800  for  5  yr.  2  mo.  5  da.       1898     1       5 

at  S%  =  $331. 56.  525 

The  first  year's  interest  is  due  Jan.  5,  1899,  and  bears 
interest  until  March  10,  1903.  The  interest  falling  due 
at  the  end  of  the  second  year  bears  interest  until  March 
10,  1903,  and  so  on. 


292  ADVANCED  BOOK  OF  ARITHMETIC 

$64  bears  interest  for  4  yr.  2  mo.  5  da. 
$64  bears  interest  for  3  yr.  2  mo.  5  da. 
$64  bears  interest  for  2  yr.  2  mo.  5  da. 
$64  bears  interest  for  1  yr.  2  mo.  5  da. 

$64  bears  interest  for 2  mo.    5  da. 

Adding,  $64  bears  interest  for  10  yr.  10  mo.  25  da. 

Interest  on  $  64  for  10  yr.  10  mo.  25  da.  at  8  %  =  $55.82. 
Interest  on  $800  for  5  yr.  2  mo.  5  da.  at  8  %  =  $331.56. 
Amount  due  =  $800  +  $  331.56  +  $55.82  =  $1187.38. 

EXERCISE   160 

1.  Find  the  amount  due  April  1,  1903,  on  a  note  for 
$900,  dated  July  15,  1899,  and  bearing  interest  annually 
at  7%. 

2.  Find  interest  from  March  6,  1898,  to  Jan.  1,  1903, 
on  note  for  $1400,  interest  payable  annually  at  8  %. 

3.  Find  amount  due  Jan.  1,  1903,  on  a  note  for  $1200, 
dated  July  1,  1897,  interest  payable  annually  at  6  %. 

4.  Find  amount  due  Jan.  1,  1903,  on  note  for  $1000, 
dated  Jan.  1,  1898,  interest  payable  semiannually  at  5%. 

5.  Find  the  amount  due  at  the  end  of    5  years  on  a 
coupon  note,  interest  payable  semiannually,  if  the  face  of 
the  note  is  $700,  and  the  rate  of  interest  is  7%. 

COMPOUND  INTEREST 

Savings  banks  and  other  banks  which  give  interest  on 
deposits  add  the  interest  semiannually  or  annually  to  the 
amount  deposited.  This  interest  added  bears  interest 
until  the  next  date  of  balancing  the  depositor's  account 
book.  The  next  interest  is  added  in  the  same  way. 

Suppose  a  person  deposited  in  a  savings  bank  $250  and 
allowed  it  to  remain  there  for  10  yr.  The  question  might 


COMPOUND  INTEREST  293 

be  asked,  how  much  will  principal  and  interest  amount  to 
at  the  end  of  that  time  ?  The  accruing  interest  in  this 
case  is  called  compound  interest. 

Designate  principal  by  p,  rate  of  interest  by  r,  time 
in  years  by  w,  and  amount  by  a. 

When  the  rate  of  interest  on  $1  is  $r  per  year,  the 
amount  of  $1  at  the  end  of  1  yr.  is  $(1  +  r). 
.-.  the  amount  of  1  1  in  1  yr.  =  f  (1-fr), 
/.  the  amount  of  $p  in  1  yr.  =$p(l  +  r). 

To  get  the  amount  of  any  principal  for  1  yr.  or  for 
6  mo.,  or  for  any  other  period  at  simple  interest,  mul- 
tiply the  principal  by  the  amount  of  $1  for  that  time. 

Take  now  $  j?(l-fr)  as  principal,  then  the  amount  of 

r)  in  1  yr.  =  $^1  +  r)(l  +  r)  =  $X1  +  O2- 
The  amount  of  $p(l  +  r)2  in  1  yr.  =  $p(l  +  r)2(l  +  r) 


Similarly,  the  amount  of  $p  (1  +  r)3  in  1  yr.  =  $  jt?(l  +  r)4. 
At  compound  interest, 

$p  amounts  in    2  yr.  to  |jp(l  +  r)2. 
$p  amounts  in    3  yr.  to 
$p  amounts  in    4  yr.  to 
$jp  amounts  in    5  yr.  to 
Generally,  $p  amounts  in  nyr.  to  $/>(! 
At  compound  interest, 

The  amount  of  $1  at  4%  for  20  yr.  is  f  (1.04)20. 
The  amount  of  $1  at  5%  for  20  yr.  is  9(1.05)*. 
The  amount  of  91  at  6%  for  20  yr.  is  f  (1.06)20. 
The    labor    of    raising    1.06    to    the    20th    power    is 
considerable. 

A  student  who  has  a  working  knowledge  of  logarithms 
can  do  this  by  the  aid  of  a  table  of  logarithms  in  less  than 
a  minute. 


294 


ADVANCED   BOOK  OF  ARITHMETIC 


The  following  table  gives  the  amount  of 
pound  interest : 


at  com- 


YRS. 

2% 

2*% 

3% 

4% 

5% 

6% 

1 

1.020000 

1.025000 

1.030000 

1.040000 

1.050000 

1.060000 

2 

1.040400 

1.050625 

1.060900 

1.081600 

1.102500 

1.123600 

3 

1.061208 

1.076891 

1.092727 

1.124864 

1.157625 

1.191016 

4 

1.082432 

1.103813 

1.125509 

1.169859 

1.215506 

1.262477 

5 

1.104081 

1.131408 

1.159274 

1.216653 

1.276282 

1.338226 

6 

1.126162 

1.159693 

1.194052 

1.265319 

1.340096 

1.418519 

7 

1.148686 

1.188686 

1.229874 

1.315932 

1.407100 

1.503630 

8 

1.171659 

1.218403 

1.266770 

1.368569 

1.477455 

1.593848 

9 

1.195093 

1.248863 

1.304773 

1.423312 

1.551328 

1.689479 

10 

1.218994 

1.280085 

1.343916 

1.480244 

1.628895 

1.790848 

11 

1.243374 

1.312087 

1.384234 

1.539454 

1.710339 

1.898299 

12 

1.268242 

1.344889 

1.425761 

1.601032 

1.795856 

2.012196 

13 

1.293607 

1.378511 

1.468534 

1.665073 

1.885649 

2.132928 

14 

1.319479 

1.412974 

1.512590 

1.731676 

1.979932 

2.260904 

15 

1.345868 

1.448298 

1.557967 

1.800943 

2.078928 

2.396558 

16 

1.372786 

1.484506 

1.604706 

1.872981 

2.182875 

2.540352 

17 

1.400241 

1.521618 

1.652847 

1.947900 

2.292018 

2.692773 

18 

1.428246 

1.559659 

1.702433 

2.025817 

2.406619 

2.854339 

19 

1.456811 

1.598650 

1.753506 

2.106849 

2.526950 

3.025599 

20 

1.485947 

1.638616 

1.806111 

2.191123 

2.653298 

3.207136 

Example  l.  Find  the  amount  of  12500  at  compound 
interest  for  12  yr.  at  5  % . 

SOLUTION,  a  =  $2500  (1.05)12  =  $2500  x  1.T95856  = 
$4489.64. 

Example  2.  Find  the  amount  of  $5000  for  8  yr.  at  4  % 
compound  interest,  the  interest  being  compounded  semi- 
annually. 

SOLUTION.  $5000  x  (1.02)16  =  $5000  x  1.372786  = 
$6863.93. 


\ 

MISCELLANEOUS  TOPICS  295 

EXERCISE  161 

With  the  aid  of  the  above  table,  find  the  amount  at 
compound  interest  of : 

1.  $2000  for  8  yr.at  4%.       3.  $5000  for  12  yr.  at  4%. 

2.  $3000  for  10  yr.  at  3%.      4.  $6000  for  10  yr.  at  5%. 

5.  $8000  for  8  yr.   at  5%,  interest  compounded  semi- 
annually. 

6.  $2250  for  6  yr.  at;  4%,  interest  compounded  semi- 
annually. 

MISCELLANEOUS  TOPICS 
WORK  AND  TIME 

Example  l.  A  can  do  a  piece  of  work  in  6  da.,  and  B 
can  do  the  same  piece  of  work  in  8  da.  In  what  time  can 
A  and  B  do  the  work  together? 

ANALYTICAL  SOLUTION.  A  does  l  of  the  work  in  1  da. 
B  does  |  of  the  work  in  1  da. 

.-.  A  and  B  together  do  (l  -f  |)  of  the  work  in  1  da. 

/.  A  and  B  do  ^  of  the  work  in  1  da. 

.*.  A  and  B  do  ^  of  the  work  in  ^  of  1  da. 

.-.  A  and  B  do  ||  of  the  work  in  ^  of  1  da. 

.•.  A  and  B  do  the  work  in  3^  da. 

Example  2.  A  cistern  has  three  pipes.  The  first  pipe 
fills  the  cistern  in  12  hr.,  the  second,  in  15  hr.,  and  the 
third  empties  it  in  10  hr.  In  what  time  will  the  cistern 
be  filled,  if  all  three  pipes  run  together,  and  the  cistern  is 
empty  when  the  pipes  start  running? 

ANALYTICAL  SOLUTION.  The  first  pipe  fills  -^  of  the 
cistern  in  1  hr. 

The  second  pipe  fills  ^  of  the  cistern  in  1  hr. 

The  third  pipe  empties  ^  of  the  cistern  in  1  hr. 


296  ADVANCED  BOOK  OF  ARITHMETIC 

/.  the  three  pipes  fill  CiV  +  T5~~lV)  °^  ^ne  cistern  in 
Ihr. 

.*.   the  three  pipes  fill  ^  of  the  cistern  in  1  hr. 

/.   the  three  pipes  fill  |§-  of  the  cistern  in  20  hr. 

.*.  the  cistern  is  filled  in  20  hr. 

Example  3.  A  and  B  do  a  piece  of  work  in  3^  hr. ; 
A  alone,  in  7  hr.  In  what  time  does  B  do  the  work? 

SOLUTION.     A  and  B  together  do  — -  of  the  work  in 

1  hr. ;  i.e.  A  and  B  together  do  -|J  of  the  work  in  1  hr. 
But  A  alone  does  ^  of  the  work  in  1  hr. 
/.   B  alone  does  (^|  —  j-)  of  the  work  in  1  hr. 
/.   B  alone  does  -|  of  the  work  in  1  hr. 
/.   B  does  the  work  in  8  hr. 

Example  4.  A  and  B  can  mow  a  field  in  21  hr.  A  can 
do  |  as  much  work  as  B.  Find  the  time  in  which  each 
4  <,  does  the  work. 

SOLUTION.  Represent  B's  work 
by  the  rectangle  mnst. 

Represent  A's  work  by  the  rec- 
tangle pqxy. 

A  will  do  in  J  hr.  the  work  A 
and  B  do  together  in  1  hr. 

/.  A  will  do  in  Jhr.  x  21  the 
work  A  and  B  do  together  in 
21  hr. 

.'.  A  will  do  in  49  hr.  the  work  A  and  B  do  together 
in  21  hr. 

B  will  do  in  |  hr.  the  work  A  and  B  do  together  in  1 
hr. 

.*.   B  will  do  in  |  hr.  x  21  the  work  A  and  B  do  together 

in  21  hr. 


m 
V 

/t-k 

a? 

MISCELLANEOUS  TOPICS  297 

/.  B  will  do  in  36 1  hr.  the  work  A  and  B  do  together  in 
21  hr. 

A's  time,  49  hr.;  B's  time,  36|  hr. 

EXERCISE  162 

1.  If  a  person  can  do  a  piece  of  work  in  7  da.,  what  is 
his  day's  work  ?     If  he  can  do  the  work  in  4^  da.,  what  is 
his  day's  work  ? 

2.  If  B  can  copy  a  manuscript  in  5|hr.,  how  much  can 
he  copy  in  1  hr.  ? 

3.  John  travels  T5g  of  the  distance  between  two  cities 
in  1  hr.     How  many  hours  will  it  take  him  to  travel  the 
remainder  of  the  distance  ? 

4.  A  can  do  a  piece  of  work  in  3  hr.,  B  in  4  hr.,  and 
C  in  5  hr.     How  long  will  it   take    the    three   working 
together  to  do  the  work  ? 

5.  A  can  do  a  piece  of  work  in  4  hr.,  B  in  5  hr.,  and  A, 
B,  and  C  together  in  1J  hr.     How  long  would  it  take  C 
alone  to  do  the  work  ? 

6.  A,  B,  and  C  can  do  a  piece  of  work  in  6,  8,  and  10  da., 
respectively.      If   they   begin   the   work    together,    what 
part  of  the  work  remains  to  be  done  at  the  end  of  the 
second  day  ? 

7.  A,  B,  and  C  can  build  a  fence  in  10, 15,  and  20  hr.  re- 
spectively.     They  work  together  for  4  hr.,  when  B  quits. 
In  what  time  can  A  and  C  finish  the  work  ? 

8.  A  cistern  has  two  pipes.     One  can  fill  it  in  20  min., 
and  the  other  can  empty  it  in  30  min.     If  the  cistern  is 
empty,  in  what  time  can  it  be  filled,  if  both  pipes  begin  to 
flow  at  the  same  instant  ? 


298  ADVANCED   BOOK  OF   ARITHMETIC 


MOTION  IN  THE  SAME  DIRECTION,   OR  IN  OPPOSITE 
DIRECTIONS 

Example  1.  A  starts  to  overtake  B,  who  is  100  yd.  ahead 
of  him.  A  travels  11  yd.  to  B's  9  yd.  How  far  must  A 
travel  in  order  to  overtake  B  ? 

SOLUTION.     A  gains  on  B  2  yd.  in  every  11  yd.  he  goes. 

.'.  A  gains  on  B  1  yd.  in  every  5J  yd.  he  goes. 

.*.  A  gains  on  B  100  yd.  in  every  550  yd.  he  goes. 

Ans.  550  yd. 

Example  2.  A  freight  train  moving  at  the  rate  of  18 
mi.  an  hour  is  78  mi.  ahead  of  a  passenger  train  moving  in 
the  same  direction  at  the  rate  of  80  mi.  an  hour.  Find 
the  distance  the  passenger  train  must  run  to  overtake  the 
freight  train. 

SOLUTION.  In  1  hr.  the  passenger  train  gains  on  the 
freight  (30  -  18)  mi.,  i.e.  12  mi. 

.*.  in  (78  -f- 12)  hr.  the  passenger  train  will  overtake  the 
freight  train,  i.e.  in  6|  hr. 

In  6^  hr.  the  passenger  train  goes  30  mi.  x  6-|-  =  195  mi. 

EXERCISE   163 

1.  Two  ships  leave  New  York  for  Glasgow,  one  on  Mon- 
day morning  at  9  o'clock,  and  the  other  on  the  following 
morning  at  9  o'clock.     Their. rates  are  15  and  21  miles 
an  hour  respectively.     How  far  from  New  York  City  will 
the  second  ship  overtake  the  first  ? 

2.  Dallas  and  Galveston  are  315  mi.    apart.     A  train 
leaves  Dallas  for  Galveston  at  8  o'clock  A.M.  at  the  rate  of 
30  mi.  an  hour.      At  the  same  time  a  train  leaves  Galves- 
ton for  Dallas  at  the  rate  of  33  mi.  an  hour.     How  far 
will  the  trains  be  from  Dallas  when  they  meet  ? 


CIRCULAR   MOTION:   CLOCKS 


299 


3.  Paris,  Texas,  is  584  mi.  from  St.  Louis.     A  passen- 
ger train  leaves  Paris  for  St.  Louis  at  6.50  P.M.     Three 
hours  later  a  freight  train  leaves    St.    Louis    for    Paris. 
When  and  where  will  they  meet,  the  rates  being  respec- 
tively 24  mi.  and  16  mi.  per  hour  ? 

4.  A  man  walking    at  the   rate  of  4  mi.   an    hour   is 
overtaken  by  a  train  88  yd.  long,  and  is  passed  in  10  sec. 
Find  the  rate  of  the  train. 

5.  A  train  going  at  the  rate  of  40  mi.  an  hour  passes 
in  6  sec.  a  man  walking  in  the  same  direction  at  the  rate 
of  4  mi.  an  hour.     What  is  the  length  of  the  train? 

6.  Two  trains  start  from  the  same  station  and  travel 
in  the  same  direction.     The  first  train  leaves  at  7  A.M., 
and  the  second  train  at  9  A.M.     How  many  miles  from 
the  station  will  the  second  train  overtake  the  first  if  the 
rate  of  the  first  train  is  30  mi.  per  hour  and  the  rate  of  the 
second  train  is  45  mi.  per  hour? 

CIRCULAR  MOTION:   CLOCKS 

Example  l.    At  what  time  between  4  and  5  o'clock  are 
the  hands  of  a  clock  together? 

SOLUTION.  At  4  o'clock  the 
hour  hand  is  20  minute  spaces 
ahead  of  the  minute  hand. 

In  1  hr.  the  minute  hand  goes 
60  minute  spaces. 

In  1  hr.  the  hour  hand  goes  5 
minute  spaces. 

.-.  ratio  of  rates  of  motion  of 
minute  hand  and  of  hour  hand  is 
60  :  5,  or  12  :  1. 


300 


ADVANCED  BOOK  OF  ARITHMETIC 


.*.  in  12  min.  the  minute  hand  gains  on  the  hour  hand 
11  minute  spaces. 

.'.in  IT*J  min.  the  minute  hand  gains  on  the  hour  hand 
1  minute  space. 

.'.in  l^y  min.  x  20  the  minute  hand  gains  on  the  hour 
hand  20  minute  spaces. 

But  IJy  min.   x  20  =  21^  min. 

/.  the  hands  are  together  21y9^  min.  after  4  o'clock. 

Example  2.  At  what  time  after  7  o'clock  do  the  hour 
and  minute  hands  first  point  in 
opposite  directions  ? 

SOLUTION.  They  will  point  in  op- 
posite directions  whenever  they  are 
30  minute  spaces  apart. 

At  7  o'clock  the  hour  hand  is  35 
minute  spaces  ahead  of  the  minute 
hand. 

.'.  as  soon  as  the  minute  hand 
gains  on  the  hour  hand  5  minute  spaces,  they  will  point  in 
opposite  directions. 

But  the  minute  hand  gains  on  the  hour  hand  5  minute 
spaces  in  l^y  min.  x  5,  i.e.  in  5T6T 
min.     (See     Example     1.)     Ans. 
5^\  min.  past  7  o'clock. 

Example  3.  When,  between  5 
and  6  o'clock,  will  the  hour  and 
minute  hands  be  at  right  angles  ? 

SOLUTION.  The  hour  and  min- 
ute hands  will  be  at  right  angles 
when  they  are  15  minute  spaces 
apart.  FlG-  i- 

/.  they  will  be  at  right  angles  between  5  and  6  o'clock 


CIRCULAR  MOTION:  CLOCKS 


301 


FIG.  2. 


when  the  minute  hand  gains  on  the  hour  hand  (25  —  15) 
minute  spaces,  i.e.  10  minute  spaces. 

l^min.  x  10  =  1014  min.  Ans. 
101-J-  min.  past  5  o'clock. 

They  will  also  be  at  right  angles 
when  the  minute  hand  gains  on  the 
hour  hand  (25  +  15)  minute  spaces, 
i.e.  40  minute  spaces.  (See  Fig.  2.) 
l^Y  min.  x  40  =  43TrT  min. 

the  hands  will  be  at  right 
angles  to  each  other  at  10^  min. 
past  5  o'clock  and  at  43^  min.  past 
5  o'clock. 

To  the  young  learner  the  following  suggestions  may  be 
of  use : 

To  solve  a  question  in  circular  motion :  first,  draw  a  dia- 
gram showing  the  things  that  move  ;  second,  find  the 
ratios  of  the  rates  of  motion  of  the  things  moving ;  third, 
having  found  the  relative  rates  of  motion  of  the  objects, 
proceed  as  in  a  simple  exercise  involving  motion  in  a 
straight  line. 

EXERCISE   164 

1.  What  angle  do  the  hour  and  minute  hands  of  a  clock 
make  with  each  other  at  1  o'clock  ?  at  2  o'clock  ?  at  3 
o'clock?  at  4  o'clock?  at  5  o'clock?  at  6  o'clock?  at  12 
o'clock  ? 

2.  What  angle  do  the  hour  and  minute  hands  of  a  clock 
make  when  they  point  to  positions  8  minute  spaces  apart? 
12  minute   spaces   apart?     19  minute  spaces  apart?    23 
minute   spaces   apart?      27    minute   spaces    apart?      50 
minute  spaces  apart?      At  what  time  between  1  and  2 
o'clock  do  the  hands  of  a  clock  make  an  angle  of  36  °  ? 


302  ADVANCED  BOOK  OF  ARITHMETIC 

3.  How  many  minute  spaces  apart  do  the  hands  of  a 
watch  indicate  when  they  make  an  angle  of  36°?    66°? 
84°?  114°?  126°?  144°?  162°?  174°? 

4.  Find  at  what  time  between  the  hours  of  2  and  3 
o'clock  the  hands  of  a  clock  are  together  ;  between  3  and 
4  o'clock  ;    between  5  and  6  o'clock  ;    between  7  and  8 
o'clock;  9  and  10  o'clock ;  10  and  11  o'clock;  11  and  12 
o'clock. 

5.  At  what  time  do  the  hands  of  a  watch  point   in 
opposite  directions  between 

(a)  1  and  2  o'clock?  (cT>  9  and  10  o'clock? 

(5)  3  and  4  o'clock?  (V)  11  and  12  oclock? 

0)  8  and  9  o'clock  ?  (/)  12  and  1  o'clock  ? 

6.  At  what  time  after  3  o'clock  do  the  hands  of  a  watch 
first  point  in  opposite  directions  ?     When  after  6  o'clock 
do  they  first  point  in  opposite  directions?     When  after 
10  o'clock  ? 

7.  At  what  time  or  times  are  the  hands  of  a  watch  at 
right  angles  between  (a)  2  and  3  o'clock?  (6)  3  and  4 
o'clock  ?  (<?)  4  and  5  o'clock  ?  (d)  6  and  7  o'clock  ?  0)  8 
and  9  o'clock?  (/)  9  and  10  o'clock  ?  (#)  11  and  12 o'clock? 
Qi)  12  and  1  o'clock  ? 

8.  At  what  time  after  4  o'clock  do  the  hands  of  a  watch 
first  point  to  positions  8  minute  spaces  apart  ?  35  minute 
spaces  apart  ?    24  minute  spaces  apart  ?    50  minute  spaces 
apart  ? 

9.  Find  the  time,  between  5  and  6  o'clock,  when  the 
minute  hand   is   ^   of  the  circumference  of  the  dial  in 
advance  of  the  hour  hand.     The  time  between  9  and  10 
o'clock  when  the  minute  hand  is  \  of  the  circumference  in 
advance  of  the  hour  hand. 


MISCELLANEOUS   EXAMPLES 


(ARRANGED  BY  TOPICS) 
NOTATION 

1.  Write  in  figures  ten  million  ten. 

2.  Write   in   figures  seven  million   two  hundred  five 
thousand. 

3.  Write  in  figures  one  billion  one  million  one. 

4.  Write  in  figures  five  million  fifty  thousand. 

5.  Write  in  figures  ten  billion  ten  million  one  hundred. 

6.  Write  decimally  :  Twenty-five  tenths.    One  hundred 
twenty  hundredths.       One   hundred  and  fifty-five   hun- 
dredths.     Ten  thousand  ten  hundred  thousandths.     One 
million  one  ten-millionths.     Fifty-five  thousand  two  hun- 
dred eighteen  ten-thousandths.     One  million  two  hundred 
thousand  four  ten-millionths.     Five  and  five  hundredths. 
Six  hundred  six  hundredths.     Seven  hundred  ten  thou- 
sandths.    Nine  hundred  and  fifteen  thousandths.     Nine 
million  and  nine  millionths. 

ADDITION  AND   SUBTRACTION 

7.  The  following  table  of  railroad  mileage  in  the  United 
States  is  taken  from  the  report  of  the  Interstate  Commerce 
Commission : 


YEAR 

MILEAGE 

YEAR 

MILEAGE 

1890 

163,597.05 

1898 

186,396.32 

1891 

168,402.74 

1899 

189,294.66 

1892 

171,563.52 

1900 

193,345.78 

1893 

176,461.07 

1901 

197,237.44 

1894 

178,708.55 

1902 

202,471.85 

1895 

180,657.47 

1903 

207,977.22 

1896 

182,776.63 

1904 

213,904.34 

1897 

184,428.47 

1905 

218,101.04 

303 


304  ADVANCED  BOOK  OF  ARITHMETIC 

Find  the  number  of  miles  built  during  each  year,  begin- 
ning with  1860,  up  to  1905. 

8.  From  9000  take  .009. 

9.  From  275  take  .000275. 

10.  Add:  657,987,324,011 

119,008,675,987 
199,887,564,999 
999,555,777,888 
987,345,234,876 
985,234,678,100 
923,524,896,987 
987,567,342,959 
725,926,846,368 
929,935,829,349 
929,563,768,968 

MULTIPLICATION 

11.  Multiply  10,500  by  60,600. 

12.  Multiply  15,010  by  50,080. 

13.  Multiply  1.01  by  7.07. 

14.  Multiply  9010.9  by  90.4. 

15.  A  train  travels  at  the  rate  of  30.25  mi.  an  hour. 
How  far  will  it  go  in  5£  hr.  ? 

16.  Find  the  price  of  87^  A.  of  land  at  $43.75  an  acre. 

17.  Find  the  price  of  4225  bu.  of  wheat  at  84^  per 
bushel. 

18.  Find  the  value  of  .1  x  .2  x  .3  x  .4  x  .5  x  .6  x  .7. 

19.  Find  the  value  of  (1.03)4. 

20.  The  number  of  bales  of  cotton  produced  in  Texas 
in  1901-02   was   2,993,000,   and   in   1900-01,    3,550,000. 
Allowing  500  Ib.  to  a  bale,  how  many  more  pounds  of 
cotton  were  produced  in  the  latter  year  than  in  the  former? 


MISCELLANEOUS  EXAMPLES  305 

DIVISION 

21.  A  steamer's  cargo  consisted  of  120,000  bu.  of  corn, 
valued  at  $57,000;  13,541  bbl.  of  flour,  valued  at  $47,499; 
3050  bales  of  cotton,  valued  at  $158,224.     Find  the  value 
of  a  bushel  of  corn,  a  barrel  of  flour,  and  a  bale  of  cotton. 

22.  Divide  26.78508  by  .072  (not  by  long  division). 

23.  The  annual  consumption  of  sugar  in  a  certain  state 
was,  in  1890,  702,201  T.,  which  was  found  to  be  49.93  Ib. 
per  head  of  population.     Find  the  population. 

24.  Make  a  column  of  eight  numbers,  the  first  of  which 
is  73,214,  the  second  is  f  of  the  first,  the  third  is  f  of  the 
second,  and  so  on  for  the  other  numbers. 

25.  How  many  miles  are  in  278,784,000  ft.  ? 

26.  Divide  1.1252- (.784)2  by  1.125 -.784. 

27.  Divide  (.75)3  -  (.26)3  by  .75  -  .26. 

28.  Divide  14.302  by  83.92,  correct  to  four  places. 

29.  Divide  24.619  by  56,000. 

30.  The  length  of  a  degree  on  the  earth's  surface  is  ap- 
proximately 69.15  mi.     Two  places  are  on  the  same  merid- 
ian and  1000  mi.  apart.     Find,  in  degrees,  the  difference 
in  latitude. 

31.  Two  places  on  the  60th  parallel  of  latitude  are  300 
mi.  apart.     Find  the  difference  of  their  longitudes.     (1°  = 
183,085  ft.) 

32.  A  bankrupt's  liabilities  are  $47,875;  his  assets  are 
$38,650.     How  many  cents  on  the  dollar  can  he  pay? 

33.  The  product  of  two  numbers  is  642,978,  and  one  of 
the  numbers  is  5.67.     Find  the  other  number. 

34.  If  the  quotient  is  24,400,  the  remainder  is  15,  and 
the  dividend  is  6,100,015,  find  the  divisor. 


306  ADVANCED  BOOK  OF  ARITHMETIC 

35.  The  total  amount  of  money  in  circulation  in  the 
United  States  on  March  1, 1903,  was  $2,353,738,834.     The 
per  capita  circulation  in  the  United  States  on  the  same  day 
was  $29.41.     Find  the  population  of  the  United  States. 

36.  Divide  the  square  of  1001  by  77  x  169. 

37.  When  450  Ib.  of  sugar  cost  $20.25,  find  the  price 
of  84  Ib. 

38.  Find  the  value  of  a  rectangular  plot  of  ground  726 
yd.  long  and  240  yd.  wide,  at  $50  an  acre. 

39.  Find,  in  United  States  currency,  the  value  of  £79. 

40.  When  1.75  yd.  of  silk  cost  $3.85,  find  the  cost  of 
14yd. 

41.  Divide  39.328  by  .0032. 

42.  If  .  6  of  a  yard  of  cloth  cost  27  ^,  find  the  cost  of  45  yd. 

43.  Divide  1  by  1.732. 

44.  Divide  the  cube  of  11.1  by  27  times  1369. 

45.  What  is  the  ratio  of  25  A.  to  640  A.? 

G.  C.  M.   AND  L.  C.  M. 

46.  Find  the  G.  C.  M.  of  288  and  432. 

47.  Find  all  the  common  measures  of  36  and  54. 

48.  Find  the  common  divisors  of  288  and  360. 

49.  Express  1110,  777,  and  1001  as  the  products  of  prime 
numbers.     Find  their  L.  C.  M. 

50.  Find  the  G.  C.  M.  of  208,  572,  and  1326. 

51.  Find  the  L.  C.  M.  of  26,  28,  48,  70,  and  117. 

52.  Find  the  G.  C.  M.  of  625  and  2525. 

53.  Find  the  G.  C.  M.  and  L.  C.  M.  of  209,  304,  and  380. 

54.  Find  the  prime  factors  of  80,850. 

55.  Two  numbers  have  for  their  G.  C.  M.  101,  and  for 
L.  C.  M.  27,573.     Find  the  product  of  these  numbers. 


MISCELLANEOUS  EXAMPLES  307 

56.  Eesolve  61,776  into  its  prime  factors. 

57.  Two  tracts  of  land,  containing   1225  acres   and   1675 
acres,  are  divided  into  farms  each  containing  the  same  number 
of  acres.     What  is  the  largest  possible  acreage  of  each  farm  ? 

58.  Telephone  poles  are  231  ft.  apart.     What  is  the  smallest 
number  of  poles  which  will  correspond  to  an  exact  number  of 
half  miles  ? 

FRACTIONS,   DECIMALS,   AND  DENOMINATE  NUMBERS 

59.  Arrange  in  order  of  magnitude  f  ,  •£,  -^. 

60.  Find  the  difference  between  the  greatest  and  the  least  of 
the  fractions  -f  ,  -f,  -|-£,  and  \\  . 

61.  Add:   21  31  5f,  3&. 

62.  Eeduce  to  its  lowest  terms  iViV 

63.  Express  as  decimals  -||-,  ^,  -^f^. 

64.  Eeduce  to  common  fractions  .0375,  .0175,  .03125. 


65.  Simplify       l        x  15|. 

66.  Simplify  2f  of  ^|  -  (  j  of  17|  of  f  of  1|). 

4¥ 

67.  Eeduce  198  ft.  tp  the  decimal  of  1±  mi. 

68.  Eeduce  2°  30'  to  the  decimal  of  90°. 

69.  Eeduce  3  pt.  to  the  decimal  of  5  gal. 

70.  Eeduce  .375  of  16s.  Sd.  -f  f  of  15s.  6d,  to  the  decimal 
of  £5. 

71.  Show  that  if  our  calendar  had  8  leap  years  in  every 
33  yr.,  it  would  be  more  correct  than  it  now  is. 

72.  If  our  calendar  were  so  arranged  that  31  leap  years 
would  occur  in  128  yr.,  how  many  years  would  elapse  before 
the  error  would  amount  to  1  da.  ? 


308 


ADVANCED   BOOK  OF  ARITHMETIC 


73.    The  following  distances  have  been  run  by  trains  in 
the  times  indicated.     Find  in  each  case  the  rate  per  hour. 


EOUTE 

DISTANCE  IN 
MILES 

TIME 

Jersey  City  to  Oakland    .     .     . 
New  York  to  Chicago      .     .     . 
Chicago  to  New  York      .     .     . 
London  to  Aberdeen    .... 

3311 
964 
962 
539.75 
510.1 

83  hr.  45  min. 
19  hr.  57  min. 
17  hr.  45  min. 
8  hr.  32  min. 
8  hr.    1  min.    7  sec. 

Albany  to  Syracuse      .... 
Erie  to  Buffalo  Creek  .... 
Camden  to  Atlantic  City      .     . 
Liberty  Park  to  Absecon      .     . 
Berlin  to  Absecon   

147.84 
86 
58.3 
49.8 
35.6 

2  hr.  10  min. 
1  hr.  10  min.  45  sec. 
45  min.  45  sec. 
37  min.  30  sec. 
25  min.  45  sec. 

New  York  to  Philadelphia  .     . 

90 

1  hr.  17  min. 

74.  Find  the  value  of  25,000  bu.  of  oats  at  46f  ^  per 
bushel. 

75.  The  price   of  oats  in  June,   1900,  was   26^  per 
bushel,  and  in  August  it  was  21^  per  bushel.     If  a  specu- 
lator lost  f  1050  by  buying  oats  at  the  former  price  and 
selling  at  the  latter,  how  many  bushels  did  he  buy  ? 

76.  A  speculator  in  Chicago  bought  10,000  bu.  of  corn 
in   February,  1901,  at  38f  ^  per  bushel,  and  sold  it  in 
December,  1901,  at  69|^  per  bushel.     Find  his  profit.* 

77.  The  total  number  of  bales  of  cotton  exported  from 
the   United   States   for    the    season    of    1901-1902   was 
6,715,793,   valued  at  $284,779,190.     Find    the    average 
price  per  bale,  correct  to  the  cent. 

78.  The  total  number  of  farms  in  Alabama  is  223,220 ; 
the  total  acreage  of  these  is  20,685,427.     Find,  correct  to 
two  decimal  places,  the  average  number  of  acres  to  a  farm. 

*  Allow  J  $  per  bushel  brokerage  for  buying  and  for  selling. 


MISCELLANEOUS  EXAMPLES  309 

79.  The  total  sugar  production  of  California  was,  in  1902, 
356,500  T.,  valued   at   $15,500,000.     Find  the  average  price 
per  100  Ib. 

80.  According  to  the  census  of  1900,  the  number  of  persons 
employed  in  manufacturing  industries  in  Florida  was  1778, 
and   the   salaries   paid   amounted   to   $1,295,139.      Find  the 
average  salary  received  by  each  person,  correct  to  the  cent. 

81.  The  total  enrollment  in  the  elementary  and  secondary 
schools  in  the  United  States  in  1901  was  15,603,451,  and  the 
total  number  of  teachers  was  430,004.     Find  the  average  num- 
ber of  pupils  to  a  teacher. 

82.  The  total  expenditure  for  higher  education  in  Canada  in 
a  recent  year  was  $1,014,254.     This  expenditure  was  19.5^ 
per  capita  of  the  total  population.     Find  the  population  of 
Canada. 

83.  The  total  expenditure  for  higher  education  in  Germany 
in  a  recent  year  was  $  7,450,366.     The  per  capita  expenditure 
was  14.3^.     Find  the  population  of  Germany. 

The  cost  of  higher  education  in  Great  Britain  and  Ireland 
for  a  recent  year  was  given  as  $8,353,655.  The  per  capita 
expenditure  was  21.7  ^.  Find  the  population  of  Great  Britain 
and  Ireland. 

84.  The  total  amount  of  money  in  circulation  in  the  German 
Empire  is,  estimated  in  our  currency,  $1,080,100,000.     The 
per  capita  circulation  is   $19.53.     Find  the  population  of 
Germany. 

85.  Express  in  feet  .002357  of  a  mile. 

86.  Find  the  value  of  ^-  of  a  ton  -f-  %  of  a  hundredweight. 
Give  your  answer  in  pounds. 

87.  How  many  times  is  12  Ib.  8  oz.  contained  in  2  T.  ? 

88.  Light  travels  at  the  rate  of  185,000  rni.  a  second.     How 
long  does  it  take  a  ray  of  light  to  pass  from  the  earth  to  the 
moon,  a  distance  of  239,000  mi.  ? 


310  ADVANCED   BOOK  OF  ARITHMETIC 

89.  How  many  cubic  yards  of  sand  are  required  to  fill  a 
street  1^  mi.  long,  40  ft.  wide,  to  the  depth  of  5  in.  ? 

90.  Express  |-  of  a  day  as  a  decimal  of  a  common  year. 

91.  If  T5g-  of  an  acre  of  land  is  worth  $  23,  find  the  value  of 
85  A.  of  land. 

92.  If  .375  of  an  acre  of  land  is  worth  $  22,  find  the  value 
of  57  A. 

93.  Multiply   68.4  by  .0027,  and  divide  the  product  by 
f  of  .96. 

94.  Find  the  value  of  .1875  of  a  guinea  +  f  of  £  1  +  .25  of 
7s.  Sd.     Give  your  answer  in  pounds,  shillings,  and  pence. 

95.  Eeduce  12s.  6cL  to  the  decimal  of  £  4  sterling. 

96.  (a)  Divide  $1293.46  by  .00  J.     (6)  Divide  $147.32  by 
.00|.     (c)  Divide  $  3473.85  by  .OOf .    (d)  Divide  $  3295  by  .OOf . 

97.  (a)  Divide   $1456.77  by  .OOf     (6)  Divide   $3947.85 
by  .OOf. 

98.  How  many  acres  in  a  field  160  ch.  long,  40  ch.  wide  ? 

99.  A  wheel  is  12^  ft.  in  circumference.    How  many  revolu- 
tions will  it  make  in  going  6  mi.  80  rd.  ? 

100.  How  many  bushels  will  a  bin  7  ft.  by  5  ft.  and  4  ft. 
deep  hold? 

101.  Keduce  ij  to  a  fraction  having  12  for  denominator. 

102.  .08  of  a  boy's  money  is  $6.     How  much  money  has 
the  boy  ? 

103.  .875  of  a  man's  property  is  valued  at  $21,700.     What 
is  the  value  of  the  man's  property  ? 

104.  How  many  acres   are   in  a  square  field  whose   side 
is  40  rd.  ? 

105.  A  and  B  can  mow  a  field  in  7  da.     A,  B,  and  C  can 
mow  the   same   field  in  5   da.   for    $50.      What  should  C 
receive  ? 

106.  Write  decimally  three-eighths  of  one  hundredth,  and 
reduce  it  to  a  simple  decimal. 


MISCELLANEOUS  EXAMPLES  311 

107.  Reduce  |>  ^,  T\-,  and  ^  to   equivalent  fractions 
having  100  for  denominator. 

108.  Find  the  difference  between  "— -  and  — —  • 

16          .16 

109.  Reduce  16|  to  an  improper  fraction  having  16  for 
a  denominator. 

110.  A  rectangular  field  which  is  18  rd.  wide  contains 
6  A.     How  much  will  it  cost  to  fence  it  at  75^  a  rod  ? 

111.  What  decimal  of  4  ft.  2  in.  is  9  ft.  6  in.  ? 

112.  Find  the  least  fraction  which  added  to  J,  |,  ^,  ^, 
and  |-  will  make  the  sum  an  integer. 

113.  Divide  27.8  of  a  yard  by  .00125  of  a  foot. 

114.  8  cwt.  20  Ib.  of  sugar  cost  $41.42.     What  will  1 
T.  cost  at  the  same  rate  ? 

115.  Find  the  least  length  which  is  a  multiple  of  1  ft. 
3  in.,  1  ft.  8  in.,  2  ft.  1  in.,  and  2  ft.  6  in. 

116.  Twelve  tenths  of  a  number  equals  42.     Find  it. 

117.  Divide  54,218  by  64,  using  the  factors  of  64. 

118.  A  train  165  yd.  long  passes  a  telegraph  pole  in  12 
sec.     Find  the  rate  of  the  train  in  miles  per  hour. 

119.  A  city  lot  42  ft.  by  120  ft.  is  sold  for  $840.     At 
this  rate,  find  the  value  of  1  A.  of  land  in  that  city. 

120.  Find  the  greatest  number  which,  when  divided 
into  1958  and  2741,  will  give  for  remainders  8  and  11 
respectively. 

121.  By  buying  eggs  at  25^  per  dozen  and  selling  them 
at  60^  a  score,  a  dealer  makes  a  profit  of  $10.01.     How 
many  eggs  does  he  sell  ? 

122.  Reduce  •£$££$  to  its  lowest  terms. 

123.  If  a  sum  of  money  which  will  pay  A's  wages  for 
41|  da.  will  pay  B's  wages  for  55|  da.,  for  how  long  will 
it  pay  both  ? 


312  ADVANCED  BOOK  OF  ARITHMETIC 

124.  If  gold  weighs  19.3  times  as  much  as  water,  and  copper 
8.9  times  as  much  as  water,  how  much  heavier  than  water  is  an 
alloy  consisting  of  16  parts  of  gold  and  3  of  copper  ? 

125.  A  rectangular  tank  is  18  ft.  8|  in.  long,  11  ft.  3f  in. 
wide,  and  contains  41  cu.  yd.,  6  cu.  ft.,  and  34^  cu.  in.     Find 
its  depth.     Find  the  area  of  each  of  its  faces. 

126.  A  tennis  court  is  42  yd.  long  and  20  yd.  wide.     It  has 
a  walk  around  it  6  ft.  wide.     Find  the  cost  of  paving  the  walk 
at  $  1.25  per  square  yard. 

127.  Telegraph  poles  along  a  certain  railroad  are  132  ft. 
apart.    Find  the  rate  of  a  train,  in  miles  per  hour,  which  passes 
18  poles  in  24  sec. 

LONGITUDE  AND  TIME 

128.  A  train  leaves  New  York  City  at  9  A.M.,  Apr.  1, 1903, 
and  arrives  in  Carson  City,  Nev.,  in  109  hr.  15  min.     Find 
the  hour  of  the  day,  and  day  of  the  month,  Standard  time, 
that  it  reaches  its  destination. 

129.  The  time  of  mail  transit  between  Chicago  and  Santa 
Fe,  N.  M.,  is  60  hr.  55  min.     "  The  California  Limited  "  leaves 
Chicago  at  10  P.M.     At  what  time,  by  the  clocks  in  Santa  Fe, 
should  "  The  California  Limited  "  pass  Santa  Fe  ? 

130.  The  longitude  of  Cairo,  Egypt,  is  31°  21'  E.,  and  the 
longitude  of  Savannah,  Ga.,  is  81°  5'  30"  W.     Find  the  differ- 
ence in  time. 

131.  The  longitude  of  Toulon  is  5°  56'  E.     The  time  differ- 
ence between  Toulon  and  Halifax,  N.  S.,  is  4  hr.  38  min.  4  sec. 
Find  the  longitude  of  Halifax. 

132.  The  time  difference  between  Toulon  and  Point  Barrow, 
Alaska,  is  10  hr.  48  min.  44  sec.     Find  the  longitude  of  Point 
Barrow. 

133.  The  time  difference  between  Osaka  and  Point  Barrow 
is  19  hr.  26  min.  48  sec.     Find  the  longitude  of  Osaka.     (See 
previous  problem.) 


MISCELLANEOUS  EXAMPLES  313 

134.  (a)  The  difference  between  Standard  and  local  time  of 
Portland,  Me.,  is  19  min.     Find  the  longitude  of  Portland,  Me. 

(6)  The  difference  between  Standard  and  local  time  of  Fort 
Wayne,  Ind.,  is  20  min.  Find  the  longitude  of  Fort  Wayne. 

(c)  Cleveland,  0.,  uses  Central  time ;  the  difference  between 
its  local  and  Standard  time  is  33  min.  Find  the  longitude  of 
Cleveland,  0. 

PERCENTAGE 

135.  The  total  sugar  production  of  the  world  for  the  year 
1902  was  9,635,000  T.     The  amount  of  sugar  consumed  in  the 
United  States  the  same  year  was  2,372,000  T.     What  per  cent 
of  the  world's  production  was  the  amount  consumed  in  the 
United  States? 

136.  The  foreign-born  population  of  New  Orleans,  according 
to  the  census  of  1900,  was  30,325 ;  of  this  number,  1262  came 
from  England,  4428  from  France,  8733  from  Germany,  5398 
from  Ireland.     What  per  cent  of  the  foreign-born  population 
of  New  Orleans  came  from  England?    from  France?    from 
Germany  ?   from  Ireland  ? 

137.  The  number  of  Canadians  in  Detroit,  according  to  the 
census  of  1900,  was  25,400;  this  number  was  26.3%  of  the 
foreign-born  population.     Find,  correct  to  100,  the  number  of 
foreign-born  population  of  Detroit. 

138.  A  horse  is  sold,  at  a  loss  of  15%,  for  $127.50.     Find 
the  cost  of  the  horse. 

139.  By  selling  silk  at  $1.60  per  yard,  a  dealer  makes  a 
profit  of  25%.     What  would  the  selling  price  be  if  he  made  a 
profit  of  12 \%  ? 

140.  When   cloth   is   sold  for   $1.04  per  yard,  a  clothier 
makes  a  profit  of  30%.     What  would  his  profit  be  if  he  sold 
the  cloth  at  96^  per  yard  ? 

141.  A  wholesale  dealer  makes  a  profit  of  10%  on  canned 
goods.     The  retail  dealer  makes  a  profit  of  25%.     Find  the 
original  cost  of  canned  goods  which  cost  the  consumer  $11. 

2i 


314  ADVANCED  BOOK  OF  ARITHMETIC 

142.  A  coal  merchant  buys  coal  by  the  long  ton  at  $  4.50  a 
ton,  and  sells  it  at  the  rate  of  $  5  a  short  ton.     Find  his  gain 
per  cent. 

143.  How  much  water  must  be  added  to  a  25%  wine  mixture 
to  make  it  a  20%  mixture? 

144.  A  sells  goods  to  B  at  a  profit  of  20%  ;  B  sells  them  to 
C  at  a  profit  of  20%  on  his  outlay;   C  sells  them  to  D  for 
$  180,  thereby  losing  16|%.     How  much  did  the  goods  cost  A  ? 

145.  A  merchant  buys  goods  at  20%,  and  10%  off  list  price, 
and  sells  them  at  the  list  price.     Find  his  per  cent  of  gain. 

146.  When  20  Ib.  of  tea  are  sold  for  what  22^  Ib.   cost, 
what  is  the  gain  per  cent  ? 

147.  (a)  A  vessel  contains  31  gal.  of  wine  and  17  gal.  of 
water.     What  per  cent  of  the  mixture  is  wine  and  what  per 
cent  is  water  ?     (5)  How  many  gallons  of  water  must  be  added 
to  this  mixture  to  make  a  mixture  containing  60%  wine? 

148.  The  following  table  gives  the  distances  from  Atlantic 
to  Pacific  ports  by  the  present  routes : 

New  York  to  San  Francisco 13,244  mi.,  nautical 

New  York  to  Sydney 14,560  mi.,  nautical 

Charleston  to  San  Francisco 13,180  mi.,  nautical 

Charleston  to  Valparaiso 8,296  mi.,  nautical 

New  Orleans  to  San  Francisco  ....     13,644  mi.,  nautical 

New  Orleans  to  Melbourne 15,535  mi.,  nautical 

Galveston  to  San  Francisco 13,826  mi.,  nautical 

Galveston  to  Wellington 14,182  mi.,  nautical 

Liverpool  to  San  Francisco 13,844  mi.,  nautical 

Hamburg  to  Callao  . 10,702  mi.,  nautical 

Bordeaux  to  San  Francisco 13,691  mi.,  nautical 

The  following  table  gives  the  distances  from  Atlantic  to 
Pacific  ports  via  the  Panama  Canal  route : 

New  York  to  San  Francisco 5299  mi.,  nautical 

New  York  to  Sydney 9852  mi.,  nautical 

Charleston  to  San  Francisco 4898  ini.,  nautical 


MISCELLANEOUS  EXAMPLES  315 

Charleston  to  Valparaiso 4229  mi.,  nautical 

New  Orleans  to  San  Francisco      ....  4698  mi.,  nautical 

New  Orleans  to  Melbourne 9826  mi.,  nautical 

Galveston  to  San  Francisco 4800  mi.,  nautical 

Galveston  to  Wellington 8392  mi.,  nautical 

Liverpool  to  San  Francisco 8038  mi.,  nautical 

Hamburg  to  Callao 6527  mi.,  nautical 

Bordeaux  to  San  Francisco 7938  mi.,  nautical 

What  per  cents  of  the  distances  by  the  old  routes  are  saved 
by  the  Panama  Canal  route  ? 

INTEREST 

149.  Find  the  simple  interest  on  $78  for  93  da.  at  8%. 

150.  Find  the  simple  interest  on  $98  for  63  da.  at  7%. 

151.  Find  the  amount  of  $179  for  123  da.  at  6%. 

152.  Find  the  simple  interest  on  £  324  7s.  9d.  from  June  12 
to  Dec.  7  following  at  5%. 

153.  Find  the  simple  interest  on  £1169  6s.  Sd.  from  Jan. 
25  to  June  18  following  at  9%. 

154.  What  principal  will  produce  $19.50  in  1  yr.  at  6±%  ? 

155.  What  principal  will  produce  $180  interest  in  3  mo. 
at  5%  ? 

156.  What  principal  will  amount  to  $  412.50  in  7|  mo.  at  5 %  ? 

157.  What  principal  will  amount  to  $  1219  in  3  mo.  5  da. 
at  6%  ? 

158.  What  principal  will  produce  $29.17  in  5  mo.  at  7%  ? 

159.  At  what  rate  will  $1000  produce  $23.33  interest  in 
4  mo.  ? 

160.  What  principal  will  produce  75  ^  interest  in  9  da.  at  6  %  ? 

161.  Find  the  exact  interest  on  $73.15  from  June  18  to 
Aug.  1  at  1%. 

162.  A  note  for  $3500  bearing  interest  at  8%  and  dated 
Jan.  2,  1900,  was  indorsed  as  follows:   June  1,  1900,  $450; 


316  ADVANCED  BOOK  OF  ARITHMETIC 

Aug.  2,  1900,  $208;  Jan.  2,  1901,  $500;  July  7,  1901, 
Oct.  4,  1901,  $500;  Jan.  11,  1902,  $300;  Aug.  4,  1902,  $700. 
Calculate,  by  the  United  States  Rule  for  partial  payments,  the 
amount  due  on  this  note  on  Jan.  1,  1903. 

163.  A  demand  note   dated  Jan.  5,  1902,  and  drawn  for 
$575  was  paid  6  mo.  18  da.  later.     Find  the  date  of  payment 
and  the  amount  of  the  note,  the  rate  of  interest  being  7%. 

BANK  DISCOUNT 

164.  A  note  for  60  da.  is  drawn  on  Jan.  10,  1903.     Find  the 
proceeds  of  this  note,  if  its  face  is  $  150,  the  date  of  discount 
Feb.  5,  and  the  rate  6%.     (Neglect  days  of  grace.) 

165.  A  note  for  $900,  dated  Mobile,  Ala.,  Jan.  8,  1903,  and 
drawn  for  90  da.,  is  discounted  March  1.     Find  the  proceeds. 

166.  A  60-day  note  bearing  interest  at  8%,  drawn  Feb.  1, 
1903,  for  $1000,  is  discounted  Feb.  27  at  9%.     Find  the  pro- 
ceeds of  this  note. 

167.  A  demand  note  was  drawn  Oct.  1,  1902,  for  $  800,  and 
paid  5  mo.  10  da.  later.     Find  the  date  of  payment  and  the 
amount  of  the  note;  rate  of  interest,  1%. 

168.  The  proceeds  of  a  note  is  $  450  when  the  term  of  dis- 
count is  93  da.,  and  the  rate  of  interest  is  8%.     What  is  the 
maturity  value  of  the  note  ? 

MENSURATION 

169.  Find  the  area  of  a  parallelogram  if  its  base  is  100  yd. 
and  its  altitude  is  75  yd. 

170.  Find  the  area  of.  a  trapezoid  if  its  parallel  sides  are  60 
yd.  and  80  yd.,  and  its  altitude  is  50  yd. 

171.  A  tract  of  land  is  sold  for  $  3943.84 ;  the  land  cost  as 
many  dollars  per  acre  as  there  were  acres  in  the  tract.     Find 
the  cost  per  acre. 

172.  Find  the  number  of  square  yards  in  the  walls  and  ceil- 
ing of  a  room  36  by  23,  and  16  ft.  high. 


MISCELLANEOUS  EXAMPLES  317 

173.  Find  the  perimeter  of  a  square  which  contains  40  A. 

174.  A  tract  of  land  in  the  shape  of  a  rectangle  contains 
320  A. ;  its  length  is  twice  its  width.    Find  its  dimensions. 

175.  Find  the  area  of  an  equilateral  triangle  one  side  of 
which  is  100  ft. 

176.  Find  the  area  of  a  regular  hexagon  each  side  of 
which  is  50  ft. 

177.  Find  the  circumference  of  a  circle  whose  radius  is 
56.5  in. 

178.  Find  the  area  of  a  circle  if  its  diameter  is  20  in. 

179.  Find  the  surface  of  a  sphere  whose  diameter  is  20  in. 

180.  Find  the  volume  of  a  cube  one  of  whose  dimen- 
sions is  1  ft.  3  in. 

181.  The  surface  of  a  cube  is  221.0694  sq.  in.    Find  the 
length  of  one  edge  of  this  cube. 

COMPARISON  OF   PRICES 

Express : 

182.  6  francs  per  kilogram  as  dollars  per  pound  Avoir- 
dupois. 

183.  5  francs  per  meter  as  dollars  per  yard. 

184.  1.369  francs  per  liter  as  dollars  per  gallon. 

185.  9  francs  per  hektoliter  as  dollars  per  bushel. 

186.  7  marks  per  kilogram  as  dollars  per  pound  Avoidu- 
pois. 

187.  4  marks  per  meter  as  dollars  per  yard. 

188.  2  marks  per  liter  as  dollars  per  U.  S.  gallon. 

189.  9  marks  per  hektoliter  as  dollars  per  bushel. 

190.  13.785  marks  per  meter  as  dollars  per  yard, 

191.  $40  per  acre  as  francs  per  hektar. 


MISCELLANEOUS   EXAMPLES  (B) 

(TAKEN  FROM  VARIOUS  EXAMINATION  PAPERS) 

1.  What  fractional  part  of  f  of  a  gallon  is  -£-%  of  a  pint  ? 

2.  The  difference  in  time  between  two  places  is  2  hr.  33  min.  - 
Find  the  difference  in  longitude. 

3.  A  bicycle  wheel  measuring  88  in.  in  circumference  r^ust 
make  how  many  revolutions  a  minute  to  run  eighteen  mileL  ail 
hour  ? 

4.  A  coal  bin  16^-  ft.  long  and  8  ft.  9  in.  wide  must  be  how 
deep  to  contain  10  T.  of  coal,  if  one  ton  of  coal  occupy  40  cu. 
ft.  of  space  ? 

5.  Eeduce  2  yr.  21  da.  to  years  and  decimals  of  a  year. 

6.  Eeduce  .09625  bbl.  to  integers  of  lower  denominations. 

7.  Find  the  value  of  a  piece  of  land  64  ch.  by  13^  ch.  at 
$48^  an  acre. 

8.  Required  the  cost  of  18  2^  in.  plank  16  ft.  long  and 
10  in.  wide,  and  33  pieces  of  scantling  2  in.  by  4  in.  16  ft.  long, 
at  $  22  per  M,  board  measure. 

9.  The  average  yield  per  bushel  of  wheat  is  14  bu.  1  pk. 
What  will  7  bu.  3  pk.  2  qt.  yield  ? 

10.  What  is  the  difference  in  weight,  expressed  in  Avoirdu- 
pois pounds,  between  100  Ibs.  Troy  and  100  Ibs.  Avoirdupois  ? 

11.  Eeduce  to  simplest  form  3_i-i-j — 2 — . 

7-1      2  +  * 

12.  If  it  cost  $510  to  fence  a  rectangular  field  98  rd.  by 
72  rd.,  what  will  it  cost  to  fence  a  square  field  of  the  same 
area? 

13.  Express  f  as  a  decimal  fraction. 

318 


MISCELLANEOUS  EXAMPLES  319 

14.  What  is  the  ratio  of  32  ft.  to  6  yd.  ?     Express  the  result 
decimally. 

15.  What  is  the  length  of  a  plank  11  in.  thick,  1  ft.  6  in. 
wide,  containing  36  board  feet  ? 

16.  When  it  is  12  M.  in  New  York  City  (74°  W.),  what  is 
the  time  in  Manila  (120°  E.)  ? 

17.  If  the  value  of  |  of  f  of  an  estate  is  $  4500,  what  is  the 
value  of  T\  of  ^  of  it  ? 

18.  At  $  16  per  M,  board  measure,  find  the  cost  of  20  plank 
2  in.  by  8  in.  18  ft.  long,  and  30  plank  1-J  in.  by  6  in.  10  ft. 
long. 

19.  A  can  do  a  piece  of  work  in  6  da.  and  B  can  do  the  same 
work  in  8  da.     How  long  will  it  take  B  to  finish  after  they 
have  worked  together  two  days  ? 

20.  20f  is  the  product  of  three  factors.     Two  of  these  factors 
are  If  and  4|.     Find  the  other  factor. 

21.  How  many  bushels  of  wheat  will  a  box  6  ft.  by  3±  ft.  by 
2  ft.  8  in.  hold  ? 

22.  How  many  yards  are  in  .04675  mi.  ? 

23.  If  the  dividend  is  807  and  the  quotient  34^,  what  is  the 
divisor  ? 

24.  How  many  rods  of  fence  will   inclose  a  square  field 
whose  area  is  20  acres  ? 

25.  Coal  sells  at  $  5.75  per  ton.     What  will  be  the  cost  of 
2315  Ib.  at  this  rate  ? 

26.  How  many  gallons  of  water  will  a  tank  5  ft.  by  2  ft.  by 
2  ft.  hold  ? 

27.  What  is  the  length  of  one  side  of  a  square  piece  of  land 
whose  area  is  538,756  sq.  rd.  ? 

28.  A  room  is  27  ft.  by  22  ft.  6  in.     How  many  yards  of 
carpet  27  in.  wide  will  be  required  to  carpet  this  room  ? 

29.  A  man  is  hired  to  dig  a  cellar  20  ft.  by  15  ft.  by  5  ft. 
How  much  money  will  he  receive  at  30^  per  cu.  yd.  ? 


320  ADVANCED  BOOK  OF  ARITHMETIC 

30.  How  many  days  are  there  between  Aug.  14  and  Dec.  29  ? 

31.  Find  the  value  of  a  car  load  of  wheat,  estimated  at 
21,643  lb.,  at  92^  per  bushel. 

32.  Two  persons  travel  in  opposite  directions  from  the  same 
point  at  the  rate  of  4^  and  7f  mi.  per  hour,  respectively.    How 
far  apart  are  they  after  traveling  37  J  hrs.  ? 

33.  A  man  was  born  Nov.  22, 1861.     What  is  his  age  to-day  ? 

34.  Factor  the  following  numbers  and  from  these  factors 
determine  the  G.  C.  M. :  42,  112,  140,  308. 

35.  What  will  75  boards  2  in.  by  4  in.  by  16  ft.  long  cost  at 
$  12  per  M  board  measure  ? 

36.  160  rd.  of  fence  will  inclose  how  many  acres  in  the  form 
of  a  square  ? 

37.  The  difference  in  longitude  between  two  places  is  7°  42' 
30".     Find  the  difference  in  time. 

38.  How  wide  is  a  rectangular  field  containing  5  A.,  the 
length  of  the  field  being  7  ch.  25  1.  ? 

39.  A  pavement  is  5^  rd.  long  and  8  ft.  6  in.  wide.     What 
did  it  cost  at  $  1.40  per  sq.  yd.  ? 

40.  Three  men,  A,  B,  and  C  do  a  piece  of  work ;  A  works 
3  da.  of  5  hr.  each,  B,  2  da.  of  6  hr.  each,  and  C,  7  da.  of  3  hr. 
each.     At  the  same  rate  of  wages,  how  should  they  divide 
$  120,  the  total  amount  received  for  doing  the  work  ? 

41.  A  miller  charges  y1^-  for  toll.     How  many  bushels  of 
wheat  must  one  take  to  mill  to  get  12  bbl.  of  flour,  each  con- 
taining 196  lb.,  if  a  bushel  of  wheat  makes  40  lb.  of  flour  ? 

42.  The  annual  rainfall  in  a  certain  locality  is  30  in.     How 
many  tons  of  water  fall  on  an  acre  of  land  in  this  locality,  if  a 
cubic  foot  of  water  weighs  1000  oz.  ? 

43.  How  much  does  a  man  gain  or  lose  on  the  sale  of  two 
houses  at  $  1200  each,  if  he  gains  ^  of  the  cost  price  on  one, 
and  loses  ^  of  the  cost  price  on  the  other  ? 


MISCELLANEOUS  EXAMPLES  321 

44.  What  is  the  ratio  of  7  Ib.  Troy  weight  to  10  oz.  Avoir- 
dupois ? 

45.  The  divisor  is  357,  the  quotient  is  6f ;  what  is  the  divi- 
dend? 

46.  A  farmer  had  28  A.  of  land  left  after  selling  \  of  his 
farm  to  one  neighbor,  f  of  it  to  another,  and  f  of  the  remainder 
to  another.     How  large  was  his  farm  ? 

47.  Multiply  8.035  by  .0035,  add  3,  and  divide  the  sum  by 
.000625. 

48.  Divide  $459.25  into  three  parts  that  shall  be  to  one 
another  as  f,  f,  and  3  respectively. 

49.  When  it  is  two  o'clock  P.M.  in  Jerusalem,  what  is  the 
time  in  Cincinnati?     The  longitude  of  Jerusalem  is  35°  12' 
E.,  and  of  Cincinnati,  84°  26'  W. 

50.  Find  the  exact  number  of  days  between  Dec.  23,  1902, 
and  to-day. 

51.  A  man's  farm  is  mortgaged  for  f  of  its  cost ;  he  sells  it 
for  $6000  which  is  25%  above  its  cost.     How  much  money 
will  he  have  after  paying  the  mortgage  ? 

52.  A  note  for  $  600,  dated  Oct.  24,  1902,  and  due  in  8  mo., 
with  interest  at  6%  per  annum,  is  discounted  at  bank  Dec.  20, 
1902.     Find  the  proceeds. 

53.  A  man  sold  two  lots  each  for  $  600,  gaining  20%  on  one, 
and  losing  20  %  on  the  other.     What  was  his  gain  or  loss  ? 

54.  A  man  buys  a  book  the  list  price  of  which  is  $  7.20,  at  a 
discount  of  16|%,  and  sells  it  for  $7.50.     What  is  his  gain 
per  cent  ? 

55.  What  principal  at  interest  for  1  yr.  3  mo.  will  amount 
to  $506,  the  rate  of  interest  being  8%  per  annum? 

56.  Twelve  per  cent  of  90  is  what  per  cent  of  100  ? 

57.  At  the  following  rates  per  annum  of  simple  interest, 
what  time  is  required  for  the  accruing  interest  to  equal  the 
principal:   6%,  8%,  9^%  ? 


322  ADVANCED  BOOK  OF  ARITHMETIC 

58.  What  is  the  exact  interest  on  $10,000  from  Jan.  18, 
1903,  to  May  6,  1904,  at  3±%  ? 

59.  A  30-day  note,  without  interest,  is  discounted  at  a  bank 
at  8%  for  $350.     What  is  the  face  of  the  note  ? 

60.  Bonds  bearing  5%  interest  are  bought  at  120.     What  is 
the  rate  of  income  on  these  bonds  ? 

61.  An  agent  buys  sugar  at  4|^  per  pound ;  his  commission 
at  |-%  is  $25.     How  many  pounds  of  sugar  does  he  buy  ? 

62.  The  discount  of  a  note,  discounted  at  bank,  for  3  mo.  18 
da.  at  5%  is  $4.20.     Find  the  proceeds. 

63.  What  single  discount  is  equivalent  to  trade  discounts  of 
25%,  10%,  and  5%  on  the  list  price  of  an  article  ? 

64.  The  property  in  a  school  district  is  assessed  at  $  196,000. 
What  rate  of  taxation  would  be  required  to  provide  about 
$  800  annually  for  the  improved  maintenance  of  the  schools  ? 
What  annual  tax  would  a  man  pay  on  this  account  whose  prop- 
erty is  assessed  at  $  1200  ? 

65.  A  man  sold  two  horses  at  $  80  each.     On  one  he  gained 
20%,  on  the  other  he  lost  20%.     Find  the  gain  or  loss. 

66.  What  must  I  ask  for  an  article  worth  $36  that,  after 
falling  20%,  I  may  gain  25%  on  the  value  ? 

67.  A  school  district  advertised  for  bids  to  build  a  school- 
house,  the  lowest  bid  being  $21,049.     If  it  costs  3%  to  collect 
the  money,  how  large  a  levy  should  be  made,  supposing  29% 
of  it  to  be  non-collectible  ? 

68.  WThat  must  a  man  pay  for  4%  stock  to  get  5%  on  his 
investment  ? 

69.  If  you  buy  United  States  3's  at  110,  what  per  cent  per 
annum  would  your  investment  pay  ? 

70.  A  merchant's  expenses  average  10%  of  his  sales.     At 
what  per  cent  advance  on  cost  must  he  sell  his  goods  to  clear 
20%  profit? 


MISCELLANEOUS  EXAMPLES  323 

71.  A  merchant  sold  goods  to  the  amount  of  $  760.95,  thereby 
losing  11%.     What  did  he  pay  for  the  goods  ? 

72.  A  ship  is  insured  for  half  its  value  for  $374.     If  the 
rate  is  2f  %,  what  is  the  value  of  the  ship  ? 

73.  A  carriage  dealer  sold  16  buggies  at  $  200  each ;  on  one 
half  he  gained  10%,  and  on  the  other  half  he  lost  10%.     Find 
his  net  gain  or  loss. 

74.  What  principal  will  amount  to  $1253.86  in  2  yr.  11  mo. 
13  da.,  interest  at  5%  ? 

75.  How  do  you  find  the  rate  per  cent  per  annum  when  the 
principal,  interest,  and  time  are  given  ? 

76.  How  do  you  find  the  principal  when  the  rate  per  cent 
per  annum,  time,  and  interest  are  given  ? 

77.  How  do  you  find  the  time  when  the  principal,  rate  per 
cent  per  annum,  and  interest  are  given  ? 

78.  The  list  price  of  office  desks  is  $  15,  but  12  desks  are  sold 
for  $  126.     What  rate  of  discount  is  allowed  ? 

79.  In  a  certain  time  $650  will  amount  to  $713.05  at  6% 

simple  interest.     Find  the  time. 

80.  A  note  for  $  500,  due  in  3  months,  is  discounted  at  bank 
at  6%.     Find  the  proceeds. 

81.  2361  is  what  per  cent  of  78£  ? 

82.  A  man  insures  his  life,  paying  a  premium  of  $  28,  which 
is  at  the  rate  of  -|%  on  the  amount  of  his  insurance.     Find  the 
face  of  the  policy. 

83.  If  25%  of  the  selling  price  of  an  article  is  profit,  what 
is  the  per  cent  of  gain  on  its  cost  ? 

84.  A  man  fails  in  business;  his  assets  amount  to  $2100, 
his  liabilities  to    $  6000.     What   per  cent  will  his  creditors 
receive  ? 

85.  What  is  the  interest  on  $  475  for  1  yr.  3  mo.  24  da.  at 


324  ADVANCED  BOOK  OF  ARITHMETIC 

86.  A  man  bought  four  loads  of  hay,  each  weighing  2750  lb., 
at  $  20  per  ton ;  he  gave  in  payment  his  note,  without  interest, 
at  60  da.     What  are  the  proceeds  of  this  note,  discounted  at 
a  bank  at  6%  ? 

87.  What  per  cent  of  -J-  is  -|-  ? 

88.  An  agent's  commissions  at  5%  amount  to  $37.65.     Find 
the  amount  of  his  sales. 

89.  The  tax  on  property  assessed  at  $  8500  is  $  48.37.     What 
is  the  rate  on  $  1000  ? 

90.  Find  the  date  of  maturity  of  a  note  made  and  dated 
Sept.  11,  1902,  and  payable  90  da.  after  date. 

91.  Find  the  cost  of  87  shares  of  stock  at  76J,  brokerage  | 
per  cent. 

92.  A  New  York  sight  draft  was  sold  in  Atlanta,  Ga.,  for 
$3542,  exchange  being  at  f%  premium.     What  was  the  face 
of  the  draft  ? 

93.  What  per  cent  of  5  lb.  is  3  oz.  Avoirdupois  ? 

94.  An  agent's  commission  for  renting  a  house  is  $13.25; 
his  rate  of  commission  is  2^%.     What  is  the  yearly  rent  of  the 
house  ? 

95.  A  man  pays  a  premium  of  $  150  for  insuring  his  house 
for  -|  of  its  value ;   the  rate  of  premium  is  1^  per  cent  per 
annum.     What  is  the  value  of  the  house  ? 

96.  A  building  worth  $6000  is  insured  for  f  of  its  value 
at  75  ^  on  the  $  100.     In  case  of  the  destruction  of  the  building 
by  fire,  what  will  be  the  owner's  loss,  including  premium  ? 

97.  What  per  cent  of  1  bu.  is  3  qt.  ? 

98.  A  merchant  can  buy  flour  on  six  months'  credit  at  $  8 
per  barrel,  or  for  cash  at  $  7.50  per  barrel.     He  buys  100  bbl., 
paying  cash,  but  borrows  the  money  at  8%  to  pay  for  it.     Is 
this  better  than  to  buy  on  credit,  and  how  much  better  ? 

99.  A  man  sells  16  shares  of  bank  stock  at  127f,  brokerage 
i-%.     How  much  does  he  receive  for  his  stock  ? 


S   EXAMPLES 


325 


ocks  at  20  %   premium  and 

discount.     What  per  cent 
> 


^s  books  show  sales  during 

me  montn  ot  March  amounting  to  f  1000.  One  half  of 
his  sales  are  at  a  profit  of  25  %  on  the  cost,  and  the  other 
half  a  loss  of  16|  %  on  the  cost.  Find  the  cost  of  the 
gcods  sold  during  the  month. 

102.  A  merchant  failing  in  business  paid  his  creditors 
$3874.75,  which  was  at  the  rate  of  55^  on  every  dollar  of 
his  indebtedness.     Find  his  indebtedness. 

103.  The  list  price  of  a  mower  is  $38  ;  the  retail  dealer 
is  allowed  discounts  of  20%,  5%,  and  3%.     What  does 
he  pay  for  mowers  ?     If  the  retailer  sells  these  mowers  at 
a   profit   of   50  %,  what   does   the   farmer  pay  for  these 
mowers  ? 

104.  A  certain  stock,  selling  at  121|,  pays  a  semiannual 
dividend  of  4%.     What  is  the  rate  per  cent  per  annum 
on  an  investment  in  this  stock  ? 


TABLES 


APOTHECARIES'  WEIGHT 
20  grains  (gr.)  =  1  scruple  (3) 

3  scruples         =  1  dram  (3) 
8  drams  =  1  ounce  (  §  ) 

12  ounces  =  1  pound  (fib) 

LIQUID  MEASURE 

4  gills  (gi.)  =  1  pint  (pt.) 
2  pints          =  1  quart  (qt.) 

4  quarts        =  1  gallon  (gal.) 
31|  gallons    =  1  barrel  (bbl.) 
2  barrels      =  1  hogshead  (hhd.) 


LONG  MEASURE 

12 inches  (in.)  =1  foot  (ft.) 
3  feet  =  1  yard  (yd.) 

5^  yards         =  1  rod  (rd.) ,  or  pole 

40  rods  =  1  furlong 

8  furlongs      =1  mile  (mi.) 
TROY  WEIGHT 

24  grains  (gr.)     =  1  pennyweight 
(pwt.) 

20  pennyweights  =  1  ounce  (oz.) 

12  ounces  =1  pound  (Ib.) 


326 


ADVANCED  BOO 


DRY  MEASURE 
2  pints  (pt.) 
8  quarts 
4  pecks 


=  1  quart  (qt.) 
=  1  peck  (pk.) 
=  1  bushel  (bu.) 

NUMERICAL  MEASURE 

12  articles  =  1  dozen 

12  dozen  =  1  gross 

12  gross  =  1  great  gross 

20  articles  =  1  score 

AVOIRDUPOIS  WEIGHT 
16  drams  (dr.)   =  1  ounce  (oz.) 
16  ounces  =  1  pound  (Ib.) 

25  pounds 
100  pounds 


CIRCULAR  ivu&Asuivr, 

60  seconds  (")  =  1  minute  (') 
60  minutes  =  1  degree  (°) 
30  degrees 


12  signs 


=  1  sign  (S.) 

=  1  circle  (C.)   or 

circumference 
=  1  circumference 


20  cwt. 
2240  pounds 


=  1  quarter 

=  1  hundredweight 

(cwt.) 

=  1  ton  (T.) 
=  1  long  ton 


360  degrees 

NAUTICAL  MEASURE 
6  feet  =  1  fathom 

608  feet  =  1  cable  length 

10  cable  lengths  =  1  nautical  mile 
(6080  feet) 

The  following  denominations  are  also  used  : 

1.152  statute  miles  =  1  geographic  mile,  or  knot 

3  geographic  miles  =  1  league 

60  geographic  miles,  or  ) 

69.1  statute  miles  }  =  1  Ae^*  °f  latltude  on  a  meridlan 

360  degrees  =  the  circumference  of  the  earth 


4  inches 

=  1  hand 

SURVEYORS'  AND  LAND  MEASURE 

9  inches 

=  1  span 

7.92  inches               =  1  link  (1.) 

21.888  inches 

=  1  sacred  cubit 

25  links                    =  1  rod 

3  feet 

=  1  pace 

4  rods                       =  1  chain  (ch.) 

TIME 

MEASURE 

10  square  chains      =  1  acre 

60  seconds  (sec. 

)  =  lminute(min.) 

640  acres                   =  1  square  mile 

60  minutes 

=  1  hour  (hr.) 

625  square  links  (sq.  1.)  =  1  pole  (P.) 

24  hours 

=  1  day  (da.) 

16  poles                       =  1     square 

7  days 

=  1  week  (wk.) 

chain 

4  weeks 

=  1  lunar  month 

30  days 

=  1  commercial 

APOTHECARIES'  FLUID  MEASURE 

month 

60  minims  (m.)  =  1  fluidrachm(f  3) 

12  months 

=  1  year 

8  fluidrachms     —  1  fluidounce(f  J  ) 

365  days 

=  1  common  year 

16  fluidounces   =  1  pint  (O) 

366  days 

=  1  leap  year 

8  pints               =  1  gallon  (Cong.) 

APPENDIX  327 

CUBIC  MEASURE 

1728  cubic  inches  (cu.  in.)  —  1  cubic  foot  (cu.  ft.) 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.) 

16  cubic  feet,  or  i  ,     ,         A  ,  A  \ 

>  =  1  cord  of  wood  (cd.) 
8  cord  feet  / 

24}  cubic  feet  =  1  perch  of  stone  or  masonry  (pch.) 

SQUARE  MEASURE 

144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.) 
9  square  feet  —  1  square  yard  (sq.  yd.) 

30£  square  yards  =  1  square  rod  or  perch  (sq.  rd.  or  sq.  pch.) 

40  square  rods  =  1  square  rood  (sq.  R.) 

4  roods  =  1  acre  (A.) 

640  acres  —  1  square  mile  (sq.  mi.) 

SPANISH  LAND  MEASURE 

In  Texas,  California,  New  Mexico,  and  other  parts  of  this 
country  which  were  formerly  parts  of  the  Spanish  empire,  the 
vara,  the  unit  of  linear  measure,  is  still  used  in  connection  with 
original  grants  of  land.  In  Texas,  the  value  of  the  vara  is 
33^  in.  In  California  and  New  Mexico  it  is  usually  considered 
33  in. 

1,000,000  square  varas  =  1  labor  =  177.136  acres 
25,000,000  square  varas  =  1  league  —  4428.4  acres 
3,612,800  square  varas  —  1  square  mile  =  640  acres 
1,806,400  square  varas  =  J  square  mile  =  320  acres 
903,200  square  varas  =  J  square  mile  =  160  acres 
451,600  square  varas  =  J  square  mile  =    80  acres 
225,800  square  varas  =  TJ?  square  mile  =  40  acres 
5645  square  varas  =  1  acre 


ANSWERS 


Exercise  4.  — North  Atlantic  Division,  5,499,620,  3,843,908,  2,866,074, 
$  105,332,839.  South  Atlantic  Division,  3,521,920,  2,324,906,  1,503,917, 
$15,907,956.  South  Central  Division,  5,002,836,  3,235,121,  2,074,304, 
$19,870,733.  North  Central  Division,  7,878,448,  5,895,631,  4,188,517, 
$  107,663,687.  Western  Division,  1,125,924,  956,472,  685,444,  $24,441,012. 
Totals,  23,028,748,  16,256,038,  11,318,256,  $273,216,227. 

Exercises.  — 1.  $2,187,217.51.  2.  $2,749,532.73.  3.  $8,593,793.27. 
4.  $733,501.99.  5.  $4,330,637.88.  6.  $9,145,702.01.  7.  $773,768.16. 
8.  $6,124,815.32.  9.  $7,750,185.14.  10.  $3,429,967.  11.  $2,001,002.77. 
12.  $3,135,817.92.  13.  $10,029,526.43.  14.  $659,959.65. 

15.  $2,004,181.95.         16.    $853,917.00. 

Exercise  6.  —  1.   $  75,  $  200,  $  900,  $  1225.      2.  $  2.08,  $  4.68,  $  24.44. 

3.  100,320ft.;    401,280ft.          4.    273  da.          5.   $6084.         6.    2136  hr. 

7.  1598  mi.      8.    62,720  A.     9.    1173  mi.      10.    487,956  Ib.       11.    $1008. 
12.   85,008  cu.  in.     13.   103,680  pages.      14.   $956.80.        15.   $1736. 

Exercise  7.— 1.    9000  sq.  yd.        2.   7056  sq.  yd.       3.    102,400  sq.  rd. 

4.  11, 760  sq.  yd.     5.    22,848  sq.  yd.     6.    167,200  sq.  yd.     7.   4725  sq.  rd. 

8.  20,160ft.  9.    252  sq.  in.  10.    36  sq.  mi.  11.    432  sq.  mi. 
12.  40,480  sq.  yd.      13.   2200  sq.  yd.      14.    5332  sq.  ft.      15.    2880  sq.  in. 

16.  1296  sq.  in. 

Exercise  8.  —  1.    $639.52.      2.   $225.40.     3.    $476.94.      4.   $898.03. 

5.  $3594.51.     6.    $5455.08.      7.  $8023.18.      8.  $2136.42.     9.  $5046.93. 
10.   $2285.44.     11.  $359.31.     12.  $555.03.     13.  $239.76.     14.  $1076.28. 
15.    $1866.95.    16.  $6399.52.    17.  $7402.50.    18.  $10,986.58.    19.  $9345. 
20.    $6594.75.     21.  $6498.38.     22.  $3269.10.     23.  $93.60.    24.  $159.60. 
25.    $187.65.      26.    $322.56.      27.    $195.44.       28.   $7.02.      29.   $19.50. 
30!    $6075. 

Exercise  9. —  1.  841.75  mi.  2.  614.2  mi.  3.  809.93  mi.  4.  710.4  mi. 
5.  1401. 56  mi.  6.  1280.2  mi.  7.  754.8  mi.  8.  1032.3  mi.  9.  1191.4mi. 
10.  1,879.6  mi.  11.  1742.7  mi.  12.  1894.4  mi.  13.  2,217,648. 
14.  2,230,095.  15.  4,821,600.  16.  1,380,768.  17.  2,423,546. 

18.    1,881,250.         19.    4,167,520.        20.   6,297,900. 

329 


330  ANSWERS 

Exercise  10.— 1.  4834,  rem.  10.  2.  4408,  rem.  14.  3.  4892,  rem.  14. 
4.  4416,  rem.  2.  5.  4166,  rem.  16.  6.  2957,  rem.  23.  7.  2485,  rem.  13. 

8.  2668.       9.   2760,  rem.  32.        10.    1706,  rem.  28.        11.   1234,  rem.  39. 
12.   2380,  rem.  28.     13.    1431,  rem.  10.     14.    2519,  rem.  25.     15.    17,530, 
rem.  43.     16.    14,616,  rem.  31.     17.    14,792,  rem.  22.     18.   8261,  rem.  35. 
19.    8741,  rem.  42.      20.   3377,  rem.  42.      21.    2133,  rem.  74.      22.    8214, 
rem.  4.     23.   9701,  rem.  37.     24.    8217,  rem.  3. 

Exercise  11.  —  1.  14  Ib.  2.  29  hr.  3.  375  pk.  4.  29  hr.,  4  mi. 
rem. ;  94  mi.  5.  384  T.  6.  179  da.  7.  225  da.  8.  2083,  4  in.  rem. 

9.  52  wk.          10.    14  doz.         11.    16.          12.    37  boxes.         13.    417  bbl. 
14.   24  hr.      15.    $  125.       16.    161  sheep.       17.    159  coins.      18.    45  hogs. 
19.   245  boxes.        20.    167  bbl.         21.    19  Ib.         22.    36  horses. 

Exercise  12. —1.    42  bags.       2.    39  bbl.  3.    79  bbl.  4.   2777yd., 

28  in.  rem.          5.    241  bu.          6.   125  da.  7.   36°.  8.   364  sq.  yd. 

9.  36  gal.        10.  33  chests.        11.  325  bbl.  12.  114  hr.  13.  1000  A. 
14.    1250  sq.  mi. 

Exercise  13.  — 1.  68.  2.  69.  3.  427.  4.  5320.  5.  245. 
6.  385.  7.  198.  8.  17.  9.  2125.  10.  717  sacks.  11.  810. 

12.  563  bbl.        13.    17,503.         14.    396.        15.    39. 

Exercise  14.  — 1.  196.2.  2.  629.8.  3.  167.6.  4.  189.8.  5.289.6. 
6.  303.8.  7.  324.8.  8.  445.2.  9.  16.3.  10.  95.6.  11.  356.1. 

13.  .5,  .75,  .8,  .3125,  .24,  .21875,  .171875,  .6171875.     14.    .85,  .38,   .1025, 
.415,  .536,  .348.      15.   2875,  .425,  .45,  .74,  .206,  .02725.      16.   .333,  .1429, 
.111,  .0909,  .0833,  .0769,  .0714,  .0667,  .0588.          17.   .1852,  .4783,  .3214, 
.1290,  .4571,  .0959. 

Exercise  15.— 1.   $58.48.       2.   $136.15.       3.    $  16.35.       4.   $31.42. 
Exercise  16. —  1.   $1.       2.  42ft.        3.  5  da.       4.  10  da,       5.  60  Ib. 

6.  $24,  5  sheep.     7.    60^  on  each  kind,  47  oranges.     8.    60^.     9.  60  sec. 

10.  7ft.  6  in. 

Exercise  18.  — 1.  8.  2.  8.  3.  9.  4.  12.  5.  15.  6.  15.  7.  18* 
8.  36.  9.  12.  10.  12.  11.  7.  12.  13.  13.  19.  14.  23.  15.  24. 
16.  28.  17.  10.  18.  15.  19.  12.  20.  13.  21.  12.  22.  18.  23.  16. 
24.  21.  25.  32.  26.  36.  27.  44.  28.  63.  29.  48.  30.  54.  31.  56. 
32.  84. 

Exercise  19.— 1.   80.     2.   42.      3.    180.      4.   90.      5.   147.      6.    140. 

7.  168.      8.    108.      9.    102.       10.    144.       11.    480.      12.   420.      13.    160. 

14.  78.      15.    336.      16.   480.      17.    756.      18.   385.      19.   270.      20.   216. 
21.  375.     22.   216.     23.    180.     24.   576.      25.    180.      26.   210.      27.    210. 
28.    420.     29.    16,800.     30.    720.     31.  300.     32.   180.     33.   630.     34.  280. 
35.    144.    36.   840.     37.    600.     38.   2002.      39.    180.     40.  210.     41.   240. 
42.    420.     43.   108.    44.    720.     45.   360. 


ANSWERS  331 
Exercise  21.  — 1.  |.    2.  y.    3.  y.    4.  -V5.    5.  -*f-.    6.  -VT3-.     7.  Y- 

15.  -3T°T2-      16-    W-      17-   ^*-.      18.  V-      I9-    -T¥-     20.    ^.  21.    1J*. 

22.  -if*.  23.  I}*.  24.  *fi.  25.  -3T2^.  26.  J^.  27.  1^.  28.  *f*. 
29.  -Vr8-  30.  -3TV. 

Exercise  22.  - 1.   f ,  j§,  J|,  if  It-      2.  f ,  Jf ,  if,  If,  «•  3.   T%,  JJ, 

if,  H,  «•    4,  A,  i2,  J},  H,  jj.     5.  I?,  if,  f5,  fj,  f|.     6.  H,  if,  J},  »}, 

If  J'  if  «,  if  *f  M-     8-  18.  if  ?f  if  tt^    »-  if  if  *f  I!'  if 

Exercise  23.  —1.    J,  f  |,  |,  f,  f,  f,  f,  f.        2.    |,  f,  |,  f 
3.    f ,  f  f ,  f,  f,  f,  f  f  -         4.    f,  f,  f,  A,.  A,  Ai  A,  A-         5. 

T96,  i  &  A- 

Exercise  24.— 1.    lrV          2.    1JJ.         3.    1TV         4.   1}.  5.   If. 

6.  1J.        7.    f.        8.    1TV         9.    If.          10.   If.         11.   If.  12.    1J. 

13.  If.          14.    1J}.          15.    1}J.          16.   2-}i.          17.    1A-  18-    2A- 

19.  2^.          20.   1}}.          21.    2T%.         22.    1}§.          23.    1T%.  24.    1}}. 

25.  |f.          26.    2 A-          27.    1J}.           28.    1J}.          29.    2J.  30.    1J}. 

31.  7A-         32.    15J.          33.    8f.          34.   25^.         35.   26}J.  36.    22f 

37.  17f         38.    12J}.         39.    19^.       40.    17}.       41.   25^.  42.    14}|. 

43.  $  A-        44.    1-A  hr.        45.   f  is  largest ;  §  is  smallest. 

Exercise  25.— 1.    J.          2.   J.         3.   }.        4.   }.         5.  |.         6.   }. 

7.  ^.          8.    }.         9.    J.           10.   A-          11-  i-          12-    }•  13-    1}. 

14.  2i. 

20.  11  A- 

26.  16}. 

32.  3i|.          33. 

38.  4|. 

44.  $2}.         45. 
Exercise  26. 

7.  27.  8.  55.  9.  3.  10.  5.  11.  2.  12.  4.  13.  20. 
14.  2.  16.  24  cows.  17.  42  doz.  18.  15  da.  19.  22  hr. 
20.  200. 

Exercise  2 7.  — 1.    12.         2.   10}.          3.    5f         4.   16J.  5.   11}. 

6.    4}.            7.    11}.            8.    24}.            9.    5f    ^       10.   12}.  11.   19}. 

12.   38}.         13.    35}.          14.    47*.         15.    38J.          16.    15}.  17.    16f 

18.  22}.        19.    33}.        20.   301}.        21.   247}.        22.   152.  23.   335}. 

24.   247}.            25.   243.            26.    160A-             27.    753}.  28.    705}. 

29.  446J.  30.  166}.  31.  221|.  32.  132|.  33.  $4}. 
34.  $42}.  35.  $15.23TV  36.  52} j*.  37.  $129.76}.  38.  $31.87}. 

39.  $3.84f.      40.    $35,40,      41.   |7.81J.      42.   4840  sq.  yd.      43.   792  in. 


if.     16.  IA- 

51.   6TV     •     22.    7}. 

'.    6}.         28.   2&- 

17.   10}.           18.    12}. 
23.    7A-         24.    3}. 
29.    11}}.         30.   10&. 

19.   8f. 
25.   51. 
31.    2f 

\.    3}}.          34.    A 

35.    6  A 

36.    2A- 

37.   6A- 

4f          40.    8H- 

41.   19f 

42.  ?A. 

43.   5f 

i.    2i}.         46.    $1.15. 

1.—  1.   1.       2.    162. 

3.   4. 

4.   35.        5.    27. 

6.    6. 

332  ANSWERS 

44.    $11.90.         45.    1036  J  mi.;  2764*  mi.        46.    $7350.        47.    1792  Ib. 
copper,  448  Ib.  lead.        48.   Gain  $805,  investment  at  end  of  year  $3105. 


Exercise 

28.  —  1. 

J-         2 

•    J- 

3.    T«r. 

4.    « 

5-    iVo- 

6. 

» 

7. 

ft. 

8 

.    f. 

9.   f. 

10. 

A« 

11.   f. 

12.    f. 

13. 

A- 

14. 

i- 

15.    A- 

16.    f. 

17. 

TV 

18.   TV 

19-   rVV 

20. 

i- 

21. 

2}- 

s 

52.    3f. 

23.    f. 

24.   f. 

25.    f. 

26.    7. 

27. 

9. 

28. 

lOf. 

29.    15. 

30. 

8J. 

31. 

22. 

32. 

26H- 

33. 

1- 

34. 

14A- 

35.    64 

|.          36.    1. 

37. 

420£. 

38.    311. 

39. 

7. 

40. 

2. 

41.    272*J$.         42.    $398|f. 

43.   $35241.          44. 

$  1.9 

3|. 

45. 

$3.65f.         46.   $162.50. 

47. 

,   $52J. 

Exercise 

29.  —1 

.    A- 

2.    £ 

*• 

3.    If. 

4.     T*8- 

5. 

1- 

6. 

& 

7.  A- 

8-    «- 

9. 

10.  A- 

11.     f- 

12. 

13. 

A' 

14.    A- 

15. 

1A. 

16. 

A- 

17. 

if. 

18. 

3i. 

19. 

*V 

20.    A- 

21. 

1- 

22. 

A- 

23. 

7%. 

24.  2 

25. 

18. 

26.   A- 

27. 

A- 

28. 

49. 

29. 

T43- 

30. 

2f 

31. 

28. 

32.   A- 

33. 

iff. 

34. 

21. 

35. 

A- 

36.    ^ 

IT?- 

37. 

28. 

38.   &. 

39. 

2A- 

40 

,   30. 

41. 

A 

42. 

/f 

43. 

321. 

44.    A- 

45.    4f. 

46, 

,   28. 

47 

.  f. 

48. 

T9- 

49. 

601. 

50.    A- 

51. 

A- 

52. 

96. 

53. 

17- 

54. 

A 

55. 

671. 

56.    A- 

57 

.   4. 

58. 

49^. 

59. 

f. 

60.    7 

61. 

2. 

62.    f. 

63. 

i- 

64.    1J. 

65. 

I  Jf. 

66. 

H- 

67. 

51. 

68.   |f. 

69. 

ft.          70.   2f.          71.    A-          72. 

If 

Exercise 

30.  —  9. 

4.          10.   9. 

11 

.    16. 

12. 

5}. 

13. 

25. 

14. 

81. 

15.    9. 

16.    |. 

17. 

1.        18.    11. 

19.   li. 

20.   i 

If- 

21. 

If 

22.    JW- 

23. 

1- 

24.    5 

2A-  ' 

25. 

*^ 

26.    1 

H- 

27.    IfJ-.         28.    1TV         29.    TV         30.    1J.         31.    6.         32. 

Exercise  31.  — 1.  $4290.  2.  $2|.  3.  $216.75.  4.  $137f 
5.  $630.  6.  $575.  7.  $24.70.  8.  $68.88.  9.  $881|.  10.  $7. 
11.  $4.25.  12.  $8.41.  13.  $7.72f  14.  $5.75.  15.  $13J. 
16.  $120.  17.  108ft.  18.  72.  19.  255  T.  20.  60. 

Exercise  33.  —  1.  666.973.  2.  151.987.  3.  261.304.  4.  1943.232. 
5.  135.148.  6.  160.687.  7.  282.238.  8.  317.187.  9.  176.483. 

10.  212.728.         11.    281.308.         12.   377.077. 

Exercise  35.  — 1.  $4.32,  $13.70,  $43.20,  $10.08.  2.  $3.105,  $22.59, 
$6.064,  $5.929.  3.  $11.90,  $34.74,  $40.992,  $32.85.  4.  $10.10,  $40.50, 
$180.30,  $24.71.  5.  $430,  $693.70,  $337.80.  6.  31.5  A.,  4  A.,  30  A. 
7.  .01,  .12,  .09,  .002,  .012.  8.  502.2.  9.  2123.2.  10.  1806.42. 

11.  432.036.         12.    .426.         13.   2.625.         14.    28.56.          15.    271.9846. 
16.   .56056.      17.   249.048.      18.   .1702.      19.   .4182.      20.   .6.    21.   3.3642. 
22.    17.328.          23.    .20001.        24.    .9409.         25.    .4624.         26.    .139129. 


ANSWERS  333 

27.  .811801.  28.  .644809.  29.  .480249.  30.  .001.  31.  .027. 
32.  .064.  33.  .343.  34.  1.124864.  35.  1.191016.  36.  1.259712. 
37.  .015625.  40.  153.9384.  41.  197,061,258.  42.  15,205,344. 

43.    1110  mi.,  66,600  mi.         44.    8796  ft.,  527,760  ft.         45.    .32  in. 

Exercise  36.  — 1.   9.16125.  2.    .8625.  3.    3.227.          4.    .199. 

5.  .9128.  6.  .490875.  7.  .003.  8.  1.47433.  9.  1.174.  10.  .657. 
11.  .925.  12.  1.142875.  13.  .02468.  14.  .12201.  15.  .1638. 

16.  .7843875.         17.    .912.         18.    .9172.         19.    .03866.         20:    .07484. 
21.    .142857.         22.    1.0535.        23.   1.0446.         24.   .2478. 

Exercise  37.  — 1.   2.925.  2.    .024.  3.    8.6875.  4.   .2058. 

5.  .3596.         6.   2.62.         7.    3.725.          8.   2.531.        9.   1.35.         10.   3.36. 

11.  6.3234.  12.   .5.         13.    40.          14.    62.5.          15.    125.         16.   50. 

17.  15.        18.  50.        19.    160.  20.    12,500.  21.    8000.        22.   500. 
23.  25.        24.  .5.          25.    1000.  26.    1250.  27.    1000.          28.    50. 
29.  250.         30.    250.         31.   25.  32.   2.5.  33.    .0125.         34.   .00125. 
35.  .0004.  36.    .00666+.  37.    .125.  38.    .15.             39.   32. 
40.  .027.       41.  48.7.       42.    8.03.  43.    .0904.  44.    .708.       45.    1.012. 
46.  .01014.  47.   .517.        48.    .01054.         49.  .00377.         50.    .0365. 

Exercise  38.  — 1.  $17,762.08.  2.  $19.740.80.  3.  $18,850.01. 
4.  $30,737.16.  5.  $21,197.90.  6.  $41,931.64.  7.  $15,101.25. 
8.  $69,022.28. 

Exercise  39.  —  1.  .375,  .625,  .4375,  .5625,  .6875,  .8125,  .9376,  .1875. 
2.  .2667,  .4667,  .7333,  .8667,  .9333,  .5833,  .9167.  3.  .3,  .79,  .087, 
.0183,  .2779.  4.  .0375,  .1375,  .3625,  .95,  .926,  .6167,  .8833.  5.  .03125, 
.09375,  .15625,  .28125,  .40625,  .59375,  .96875.  6.  .2857,  .7143,  .0769, 
.3077,  .2143,  .6429,  .7857,  .9286.  7.  .5556,  .1919,  .1471,  .0544,  .0274,  .0569. 

Exercise  40.  -1.    &,  f,  J,  J,  A-      2.   TJ^,  ^  *fo»  A»  dhr-    3-   To9o<r> 

9  911  A  9  13  7  119  3  K          1 8  fi  3  48          «  1 

¥00"'  S7>    407'  *•     T25»    T2^>     ^¥tf»    T~QQ^Qi    SO'  »'     T25'    Ittfr     6^3'  5frO> 

vh-         6.    T\<V,  6%,  A3o,  *fo,  H-          7.    ||,  jj,  f  J,  U- 

Exercise  41.  — 1.   4  hr.         2.    12  hr.  3.    6.6667  hr.         5.    14  hr. 

6.  50.1875  mi.  per  hr.         8.    53.33  mi.  per  hr.         9.   54.13  mi.  per  hr. 
10.   29.13  mi.    per  hr.  11.    546.45  mi.   per  da.,   22.77   mi.    per  hr. 

12.  1233.3  times.  13.   $2400.  14.    $280.  15.   850   cu.   ft. 
16.   420  cu.  ft.       17.   7.48  gal.       18.   $38.28.      19.   $26.25.      20.   $17.55. 

Exercise  43. -1.   &.         2.   1         3.   |.      4.   TV       5.    &.        6.   f. 

7.  Iff.          8.    If.        9.    f.        10.   tV         11-   f-        12-   it-         13-    rf- 
14.   if.       15.    5,      16.   "H.       17.   1J.       18.   Iff.       19.    Jo-        20.   1|. 

Exercise  44.  —  1.  413.33  sq.  yd.  2.  149.33.  sq.  yd.  3.  1966J 
sq.  yd.  4.  6442J  sq.  yd.,  2512  ft.  5.  14,400.  6.  (a)  15,762.22  sq.  ft., 
(5)  21,112.5  sq.  ft.,  (c)  4430.54  sq.  ft.,  (d)  1291.79  sq.  ft.,  (e)  16,032.99 


334  ANSWERS 

sq.  ft.,    (f)  10,911.25   sq.  ft.,   (g)   17,459   sq.    ft.,   (ft)    80,711.38   sq.  ft. 

7.    (a)  720.6  sq.  yd.,  (ft)  849.58  sq.  yd.,  (c)  912.88  sq.  yd.,  (d)   1626.31 

sq.   yd.,    (e)    1146.23   sq.  yd.          8.    4096  bu.,  $1996.80.         9.    71. 6  A. 

10.    1425.6  thousand.  11.    8640  tiles.             12.   62.2  sq.  yd.,  83.55 
sq.  yd.          13.   21.9  T. 

Exercise  45.—  1.    $11.32.         2.    $28.57.       3.   $5.88.        4.    $41.60. 

5.  $42.16.          6.    $38.50.          7.    $30.11.          8.   $24.60.          9.    $28.56. 

10.  $41.47.        11.    $67.20.       12.    $21.80.        13.    $24.75.       14.    $39.15. 

15.  $25.85.        16.   $66.70.         17.   $35.         18.    $23.25.        19.    $37.80. 

20.  $40.50.       21.    $38.95.      22.   $4.41.         23.    $92.16.       24.   Weimer, 
$26.91;  Flatonia,  $24.76;  Columbus,  $27.30;  Beaumont,  $42.31;  Gon- 
zales,  $23.59  ;       Schulenburg,    $26.13.  25.   $31.50.         26.    $39.15. 

27.  $32.50.       28.    $10.       29.    $118.25.       30.    $113.85.       31.    $129.48. 
32.   $312.90.        33.    $435.20.         34.   $589.68. 

Exercise  46.  —  1.   $7.56.      2.   $34.       3.    $48.       4.   $100.      5.  $260. 

6.  $315.        7.    $225.       8.    $440.       9.   $870.       10.    $144.       11.    $49.15. 
12.   $60.      13.    $51.     14.    $87.50.      15.   $1210.      16.   $987.      17.    $1800. 
18.  $1625.         19.   $3000.          20.    $2450.         21.    $1960.        22.    $7800. 
23.    $512.         24.    $1656.         25.   $5000.       26.    $12,150.        27.    $262.50. 

28.  4455  girls,   3795  boys.          29.  42  A.          30.    $  1516.80.         31.    73, 
292.         32.    $223.20,    $3496.80. 

Exercise    47.  — 1.   $3200.          2.  $2160.          3.   $1230.          4.  $900. 

5.  $106.       6.    $660.       7.    $410.      8.   $1740.      9.   $3575.       10.    $470. 

11.  $3769.60.         12.   $4410.          13.    $301.          14.    $646.         15.    $1040. 

16.  $1120.  17.  $200.  18.    $784.  19.    $1500.  20.   $480. 

21.  490  trees.     22.   160,  300.     23.   922.5  A.    24.    35  ft.     25.   3542.97  mi. 

Exercise  48.  — 1.  $24,  $30,  $36,  $48.  2.  $59.50,  $68,  $76.50. 
3.  $28.50,  $38,  $76.  4.  $78.56,  $147.30,  $88.38.  5.  .$18.45,  $22.14. 

6.  $13.80,   $16.10.     7.   $20,  $24.      8.    $38.50,  $22.     9.  $25.60,  $19.20. 
10.    $32.50.         11.   $23.         12.    $67.50.        13.   $37.50.         14.   $47.50. 
15.    $93.50.          16.   $97.50.         17.    $72.         18.   $91.80.         19.    $59.40. 
20.   $120.       21.   $101.50.       22.    $136.50.       23.    $240.62.       24.    $263.20. 
25.    $106.65.        26.   3247.50.       27.    $219.       28.    $43.20.       29.    $139.50. 
30.    $136.         31.    $141.75.      32.    $123.50.       33.    $86.40.      34.    $180.50. 
35.   $210.  36.   $315.          37.    $2500.          38.   $248.          39.   $259.20. 
40.   $34.20.          41.    $30.62.          42.    $45.         43.   $150. 

Exercise  49.  — 1.  4  cwt.  87  Ib.  8  oz.  2.  6  cwt.  10  Ib.  3  oz. 
3.  2  T.  7  cwt.  18  Ib.  9  oz.  4.  31  T.  5  cwt.  5,  3  T.  19  cwt,  87  Ib. 
6.  16  T.  721  Ib.  7.  112  T.  8.  4Jf  T. 


ANSWERS  335 

• 

Exercise  50.—  1.  38,000  Ib.  2.  28,400  Ib.  3.  34,300  Ib. 

4.  803,200  oz.  5.  7502  Ib.  6.  9158  Ib.  7.  16,273  Ib.  8.  56 
short  tons.  9.  $81.  10.  700  long  tons.  11.  616  T.  12.  1497  Ib. 

Exercise  51.— 1.  168  in.  2.  3972  in.  3.  9680yd.  4.  12,980 
yd.  5.  18,209yd.  6.  784,520yd.  7.  13,622.4yd.  8.  2672  rd. 
9.  3927ft. 

Exercise  52.— 1.  102,400  sq.  rd.  2.  1600  A.  3.  522,720  sq.  ft. 
4.  7350f  sq.ft.  5.  16,665  rd.  6.  42,900ft.  7.  4840yd.  8.  6600yd. 

9.  11,220ft.         10.   675  cu.  ft.         11.    1046  cu.  ft.        12.    816,480  en.  in. 
13.    506J  cu.  ft.      14.    944,784  cu.  in.     15.    10  qt.     16.    23  qt.     17.  25  pt. 
18.    57  pt.          19.    154  pt.          20.    128  qt.          21.    124  qt.         22.   116  qt. 
23.    56  qt.         24.    59  qt.         25.    1208  pt.          26.    838  pt.        27.    2640  ft., 
1320  ft.,  480  ft.     28.   440  yd.,  352  yd.,  586f  yd.     29.   J,  TV,  ^V    30.    1210 
sq.  yd.,  2420  sq.  yd.        31.    ¥V         32.    140  sq.  yd.         33.    112  sq.  rd., 
144  sq.  rd.       34.  204r\  sq.  ft.          35.  33.6  cu.  in.          36.    28,875  cu.  in. 
37.  6  qt.        38.    93.1  gal. 

Exercise  53.  — 1.    29  gal.  1  qt.      2.    1  cu.  yd.  729  cu.  in.      3.   432  gal. 

4.  32  mi.     5.   31  bu.  1  pk.      6.   8  cu.  yd.  11  cu.  ft.  744  cu.  in.      7.  31,250 
sq.  mi.       8.  11  sq.  yd.  5  sq.  ft.  24  sq.  in.       9.    7926.59  mi.      10.    7899.58 
mi.          11.   27.01  mi.          12.   186,325  mi. 

Exercise  55.  — 1.   66,020".     2.   324,000".     3.    43,510".      4.    27,900". 

5.  433,080".  6.    163,820".  7.   855'.          8.   4545'.        9.    15,247.5'. 

10.  280'.       11.    1130'.       12.   832R        13.    525,600mm.         14.   1,578,240 
min.      15.  41,760  min.       16.  129,600  min.,  131,040  min.     17.   31,556,926 
sec.  18.  36,892,800  times.     147,268,800  times.          19.   311,760  min. 
20.    584  da.       21.   960  hr.      22.    178,826.4  hr.       23.   2922  da.       24.    48°. 
25.    174°. 

Exercise  56.  — 1.  32  yd.  2  ft.  4  in.  2.  151  yd.  9  in.  3.  25  yd.  9  in. 
4.  263  A.  87  sq.  rd.  5.  506  A.  27  sq.  rd.  6.  971  A.  79  sq.  rd. 

7.  16  gal.  3  qt.     8.   44  gal.  3  qt.  1  pt.     9.  42  bu.  3  pk.  4  qt.     10.    38  bu. 

2  pk.  3  qt.          11.   77  gal.  3  qt.         12.    153  bu.  3  pk.  5  qt.         13.    117  bu. 

3  pk.  3  qt.         14.    183  bu.  1  pk.  7  qt.      15.    147  T.  1379  Ib.       16.   667  Ib. 

4  oz.  17.   546  Ib.   13  oz.          18.   77  T.  628  Ib.          19.   272  Ib.  7  oz. 
20.   225  T.  1908  Ib.        21.   218  da.  18  hr.  30  min.       22.    241  wk.  6  da.  13 
hr.     23.    85  hr.  14  min.  52  sec.      24.    141  da.  13  hr.  47  min.       25.  38  wk. 
Oda.  12  hr.  26.   54  hr.  20  min.  3  sec.  27.   64  cu.  ft.  1032  cu.  in. 
28.   307  cu.  yd.  9  cu.  ft.         29.   362  cu.  yd.         30.   67  cu.  ft.  1023  cu.  in. 

Exercise  57.  — 1.   79°  2' 17".  2.   8°  0' 56".  3.   51°  54' 25". 

4.    117°  45'  25".          5.    6°  54'  45".          6.   66°  46'  6".          7.    85°  6'  10". 

8.  45°  57'.  9.  60°  5'  43".          10.   78°  59'  50".  11.    43°  5'  48". 


336  ANSWERS 

• 

12.  1  bu.  3  pk.  7  qt.  13.    8  gal.  0  qt.  1  pt.  14.   6  gal.  0  qt.  1  pt. 

15.  5  gal.  2  qt.  1  pt.           16.    19  bu.  2  pk.  5  qt.  17.   9  bu.  1  pk.  5  qt. 
18.  19  gal.  2  qt.  1  pt.         19.    21  gal.  0  qt.  1  pt.  20.    3  qt.       21.   28  qt. 
22.  120  qt.              23.  1  da.  20  hr.  57  min.  24.   6  da.  17  hr.  51  min. 
25.  11  hr.  40  min.  26.  2  hr. 

Exercise  58.  —  3.   4  yr.  6  mo.  28  da.  4.   6  yr.   4  mo.  29  da. 

5.  Milton,  65  yr.  10  mo.  29  da. ;  Pope,  56  yr.  0  mo.  9  da. ;  Shakespeare. 
52  yr. ;  Burke,  67  yr.  5  mo.  27  da. ;  Lee,  63  yr.  8  mo.  23  da.  ;  Grant,  63 
yr.  2  mo.  26  da.  ;  Goldsmith,  45  yr.  4  mo.  24  da.  ;  Franklin,  84  yr.  3  mo.; 
Hamilton,  47  yr.  6  mo.  1  da.  ;  Longfellow,  75  yr.  0  mo.  27  da.  ;  Newman, 
89  yr.  5  mo.  20  da. ;  Gladstone,  88  yr.  5  mo.  10  da. 

Exercise  59.  —  1.   42  yd.  2  ft.  3  in.  2.    72  yd.  1  ft.  3  in. 

3.  79  yd.  2  ft.  4  in.      4.   26  bu.  1  qt.      5.   53  bu.  1  pk.      6.    59  gal.  1  qt. 
7.  79  gal.  2  qt.  8.   27  bu.  3  pk.  1  qt.  9.   146  T.  800  Ib. 

10.  118  T.  1483  Ib.  11.    171  T.  540  Ib.  12.    87°  51' 28". 

13.  249°  38'.  14.   779  A.  40  sq.  rd.  15.   166  A.  137  sq.  rd. 

16.  254°  18' 36".        17.   11,250  Ib.         18.    75,240ft. 

Exercise  60.  — 1.   2  yd.  1  ft.  3  in.       2.    13°  19'  19f".      3.    7°  1' 111". 

4.  5  yd.  2  ft.  5i  in.  5.   11  yd.  4  in.  6.    2  bu.  1  pk.  2  qt. 

7.  10  times.      8.   240  times.      9.    1609.35.      10.    4°  5' 27T3r".      11.   264. 

Exercise  61.  — 1.   1750  Ib.       2.    9'.         3.   18  hr.  40  min.      4.    469  Ib. 

5.  22  hr.  48  min.  6.    12  qt.  7.    97,152  ft.  8.    1120  yd. 
9.  60  rd.         10.   945'.         11.    243  da.  8  hr.         12.    114  da.  1  hr.  30  min. 
13.    1  pk.  6  qt.  3.2  gi.                 14.    2  qt.  1  pt.  1J  gi.                15.    136  sq.  rd. 
16.   4290  sq.  yd.         17.    26  qt.         18.    3J  pt. 

Exercise  62.  — 1.   .8333.      2.    .16667  nearly.     3.    .06667.        4.   .375. 
5.   .4375.        6.   .5625.       7.    796,875.        8.   .09375.       9.    .195.       10.    jfc. 

11.  .25. 

Exercise  63.  — 1.   1  A.         2.    22  A.         3.   2.492  A.         4.   1493 J  A. 
.$37,3331.         5.    3.6  A.         6.    1200  A.     $58,800. 
Exercise  64.  — 3.    81200.          4.   $5000. 

Exercise  65.  — 1.    504  cu.  ft.  2.   72  cu.  yd.  3.    9216  cu.  ft. 

4.  3150  cu.  yd.  5.    13,824  gal.  6.   115.71  bu.          7.    560  cu.  yd. 

8.  3520  cu.  yd.      9.   270  boxes.      10.    1,251,655$  cu.  yd.       11.    24.48  T. 

12.  1481}  Ib.  13.    120  T.  14.  212  Ib.  |  oz.  15.    1728  boxes. 
16.  45  cd.         17.    453.75  T. 

Exercise  66. —1.  66J  sq.  ft.  2.    166f  sq.  ft.  3.   103  J  sq.ft. 

4.   143|  sq.  ft.          5.   306^  sq.  ft.  6.   301J  sq.  ft.          7.   238T\  sq.  ft. 


ANSWERS  337 

8.    177^  sq.  ft.  9.    596||  sq.  ft.       10.   1287^  sq.  ft.       11.    1346T\  sq.  ft. 

12.   708|  sq.  ft.  13.    775r72  sq.  ft.        14.   2646|  sq.  ft.        15.    76£  sq.  ft. 

16.    372  sq.  ft.  17.   3.15  A.            18.   4200  sq.  ft.            19.    247  sq.  ft. 

20.  4053  sq.  ft.  21.   2310J-f  sq.  ft. 

Exercise  67.— 1.  12.  2.  12,42^.  3.  60,60.  4.  96. 
5.  (a)  36,  (6)  48,  (c)  54,  (d)  42J,  (e)  54,  (/)  84,  (</)  87J, 
(ft)  122J.  6.  $264.  7.  $367.50.  8.  77  bd.  ft.  9.  87 J  bd.  ft. 
10.  96  bd.  ft.  11.  364£  bd.  ft.  12.  544  bd.  ft.  13.  10  cu.  ft. 

14.  396  bd.  ft.    15.  20|f  bd.  ft.    16.  141|  bd.  ft. 

Exercise  68. —  1.   81.45  perches,  44,352  bricks.  2.    1984  cu.  ft., 

1856  cu.  ft.  3.   13,200  bricks,  $118.80.  4.   1000  cu.  ft.,  198,000 

bricks.         5.    $1909.84. 

Exercise  69.  —  1.  42|  yd.  2.   64yd.   3.  24yd.    4.  66|  yd.   5.  $33.60. 

Exercise  70. —1.    24  da.          2.  114  T.          3.    10 J  hr.          4.  49£  yr. 

5.   $42331,  $16931.        6.   $200.        7.   $300.         8.   \\\.      9.    10  and  15. 

10.  i,  81  cattle.         11.   75  T.        12.    165.  13.   $340.          14.   $6300. 

15.  $1939.  16.   1600.  17.   If  hr.  18.     30  mi.,  6  hr.  56  min. 
19.  37ift.          20.    i.          21.   $2.          22.   48  oranges.  23.    41^  yd. 

24.  60^perlb.         25.    91  j*  per  yd.          26.   $158.82.         27.    42  pencils. 
28.  $43.20.          29.   27^.          30.   $1.12.  31.   40  tiles.          32.   .783. 
33.  21, 150  sec.        34.   $98.67.        35.    $750.        36.   79f  yd.       37.   $672. 
38.  $170.80.              39.   4620ft.               40.    3025  sq.  yd.  41.    240  bu. 
42.  $90.        43.    174.3768  Ib.      44.   4800  gr.       45.   48  mi.        46.   22.5  A. 
47.  $562.50.              48.   $95.               49.   June  30.              50.    1944  sq.  in. 
51.  24  sq.  rd.         52.    56.         53.   $91.80.         54.    $147.         55.    180  min. 
56.  49^  mi.           57.   $21.11.           58.    10  inin.  50  sec.            59.    345.6  Ib. 
60.  2520  gal.       61.    $45.        62.   37 J  cords.        63.    70?.       64.   345.6  bu. 
65.  .68,  .6818,  .012,  .8571.                   66.    J,  rffo,  f.                  67.   87799.3. 
68.  $500.50.          69.   5  gal.  2  qt.          70.    144  cu.  in.          71.    $60,138.75. 

Exercise  71.  — 1.  ft.  2.  11JJJ.  3.  fJJ.  4.  3J}J.  5.  1JJJ. 
6-  9JH-  7.  HJ  8-  ISftV  9.  iff.  10.  10&.  11.  8JJ. 
12.  9&V-  13.  2JJ.  14.  -ii  15.  -&V  16-  Th-  W.  A- 

1ft         5  1Q        1  Oft         1  O1  1  OO          V  O1        1  Od.       Q13 

JLO.      y2"«  Av.      g^-.  <6U.      "Jo".  <wA.      yj^*  <w<w.      1^'Q'          <6O.     ~Q~§'          *rx.*          T¥* 

25.  ff.  26.    2f|.  27.   2jf.      '     28.    f|.     '      29.    Iff.  30.   3|. 
31.   111.       32.   $1272£.       33.    16^.      34.    66 A-       35.     fffc. 

Exercise  72.  —  1.  If.  2.  3.  3.  J.  4.  1.  5.  J.  6.  1JJ. 
7.  If.  8.  8.  9.  J.  10.  20.  "  11.  1.  12.  8TV  13.  2J. 
14.  171.  15.  5.  16.  TV  17.  TV  18.  |§f  19.  ij.  20.  1^. 

21.  12T3T.         22.    15i|.        23.    1^.        24.   0.         25.   0.        26.    3|. 

z 


338  ANSWERS 

Exercise  73.  —  1.  14.  2.  1J.  3.  6T6T.  4.  ^.  5.  f.  6.  2j. 
7.  1.  8.  TV  9.  li|.  10.  ljfl.  11.  3J.  12.  Tjjfd.  13.  J. 

14.  _3_.         15.   i.         16.    31.         17.    82%.         18.   4. 

Exercise  74.  —  1.  12.88.  2.  24.75.  3.  12.78.  4.  42.72.  5.  295.2. 
6.  203.76.  7.  24.0.  8.  184.1.  9.  385.2.  10.  785.  11.  108.13. 
12.  308.4.  13.  116.  14.  1218.  15.  122  J.  16.  310.5.  17.  5750. 
18.  1713.  19.  102.6.  20.  114.975.  21.  118.3.  22.  612.72.  23.  8.45. 
24.  20.31.  25.  331%,  25%,  20%,  121%,  6J%,  15|%,  43}%.  26.  4%,  8%, 
121%,  1.65%,  i%,  |%,  .24%.  30.  $1125.  31.  11.2  Ib.  32.  27£  Ib. 
33.  $2050.  34.  N.Y.,  $476,850,000;  Boston,  f  132,982,500 ;  Phila., 
$64,387,500;  Baltimore,  $113,475,000;  N.O.,  $81,600,000;  Galveston, . 
$91,417,500. 

Exercise  75.  — 1.  3700.  2.  1248.  3.  $864.  4.  7200,6000,4500, 
4000.  5.  4000,  3000,  2400,  2000,  1500,  1000.  6.  $12,500,  $11,250, 
$10,000,  $6000.  7.  $360,  $432,  $324,  $243.  8.  $7200,  $6750, 

$5400,    $3600,   $2700.  9.    $1120,    $672,   $480.  10.    4200  oz. 

11.    76,300,712.         12.   39,007,793.         13.    37,220,331.         14.   6,166,710. 

15.  $213,293,651.      16.  $1,960,233.      17.  3500.     18.  3500.      19.  304,800. 

20.  4000  Ib. 

Exercise  76. -1.  80%.  2.  20%,  35%,  55%,  65%.  3.  5%,  20%, 
30%,  42J%.  4.  10%,  13J%,  30%,  12 J%.  5.  121%,  15%,  37.]%,  60|%. 
6.  6%,  7%,  81%.  7.  20%,  9%,  16|%.  8.  9^-%,  33*%,  15%.  9.  If  %,  8J%. 

10.  31%.         11.  2J%.         12.  f%.         13.  4^-%.       14.  6i%.       15.  12J%. 
16.25%.     17.331%.     18.  115. 15%  nearly.     19.88.44%.      20.  1^%,  10%. 

21.  6i%.         22.  20%,  60%,  80%.          23.  91.44%  nearly,  109.36%  nearly. 
24.  62.14%  nearly.      25.  82f%.      28.  109f%.     27.   (a)  1.43%,  (6)2.42%, 
(c)  .65%,  (d)  1.02%,  (e)  .71%,  (/)  .63%,  (0)  1.16%,  (ft)  .3%,  (Q  .97%. 
28.  Mobile,    23.79%;    Little   Rock,    48.05%;     Los    Angeles,     103.35%; 
Denver,   25.44%;    Pensacola,    51.04%;    Savannah,   25.6%;    Springfield, 
30.84%;   Evansville,    16.26%;    Dubuque,   19.75%;    Kansas   City,   Kan., 
3-1.19%;   Lexington,  22.27%;    Kansas  City,  Mo.,  23.38%;    Minneapolis, 
23.05%.      29.  1.12%  nearly.        30.  9.07%  nearly.      31.  64.69%,  13.12%, 
22.19%. 

Exercise  77.  — 1.  $48.  2.  $52.50.  3.  $11.25.  4.  $11.20. 
5.  $21.60.  6.  $27.20.  7.  $59.50.  8.  $65.  9.  $61.33.  10.  $50. 

11.  $133.          12.  $132.          13.  $83.60.         14.    $91.20.  15.  $106.25. 

16.  $171.50.              17.  $204.48.             18.  $679.62.  19.  $3017.60. 
20.  $2522.16.             21.  $4904.17.            22.  $2334.37.  23.  $3437.98. 
24.  $4454.66.             25.  $752.25.              26.   $319.44.  27.  $633.88. 
28.  $1891.62.      29.  $917.60.      30.  $2456.26. 


ANSWERS  339 


Exercise  78.  —  1.  $47.25.          2.  $612.          3.  30%.  4. 

5.  $283.50.       6.  $232.56.       7.  $10.        8.  32}%.       9.  24f%.       10.  25%. 

11.  $2,  $25.  12.    (a)  $241.92,  (6)  $547.20,  (c)  $405,  (d)  $739.20, 
(e)  $403.20,  (/)  $739.20,  (g)  §943.92,  (ft)  $962.50. 

Exercise  79.  — 1.  (a)  $168,  (6)  $93.75,  £c)  $33.32,  (d)  $250.83, 
(e)  $685.80,  (/)  $344.50,  (p)  $608,  (ft)  $269.10,  (i)  $514.12.  2.  (a)  25%, 
(6)  Wo,  (^  66|%,  (d)  331%,  (6)  28%. 

Exercise  80.  — 1.  (a)  $50.40,  (6)  $132,  (c)  $50,  (d}  $161.54, 
(e)  $200,  (/)  $300.  2.  (a)  $45.89,  (5)  $109.37,  (c)  $  128.57,  (d)  $91, 
(e)  $89.85,  (/)  $40.32,  (</)  $69.75,  (ft)  $289.14. 

Exercise  81.  — 1.    $3250.     2.  $300.     3.  $3640.     4.  12}%.     5.  8J%. 

6.  20%.      7.    10%.      8.   9%.     9.    $97.     10.  $86.     11.   $75.     12.   $105. 
13.    $80,  $15.     14.  $120.     15.  $4.20.     16.  25%.     17.  $982.     18.   12,080. 
19.  $16.67  nearly.    20.  $200.     21.  $10,240.    22.  $11,520.    23.  $85,700. 

24.  475  Ib.     25.   12%.      26.  14f%.      27.  22|%.      28.  $165.      29.  177$%. 
30.    5^.     31.  $2.10  per  yd.      32.  24^.      33.   16|%.      34.  38|%.     35.  84  J* 
perlb.        36.  11  J%.        37.  $98.        38.  $98.70.        39.  96} 0.        40.  440. 
41.  200.     42.  33i%.     43.  $148,  30|%.     44.  (a)  40%  above  cost,  (6)  30% 
above  cost,  (c)  60%  above  cost,  (d)  22 1%  above  cost,  (e)42f  %  above  cost. 

Exercise  82.  —  1.  $320.  2.  $1050,  $16,450.  3.  $45,  $705.  4.  $4. 
5.  96,000  bu.  6.  $56.16,  $  1815.84.  7.  $4.50.  8.  $56.70,  $  1563.30. 
9.  $1125,  $21,375,  95%.  10.  900  T.  11.  844  bales,  8^  per  pound. 

12.  $17.28.     13.  3J%.     14.  227,200  bu.     15.  $4200.     16.  $  7800,  $  7449. 
17.  $25.       18.  $235.50,78]^.       19.  4000  bu.      20.  3750  Ib.      21.   150  A. 
22.   (a)   $17.85,  $205.27;    (6)    $40.56,  $770.69;    (c)  $85.84,  $987.16; 
(d)  $28.27,  $537.23;  (e)  $100.01,  $1233.49;  (/)  $47.60,  $428.40 ;  (0)  $24, 
$276;   (7i)  $103.20,  $1960.80;  (i)  $  10.06,  $191.19.    (j)  $10.18,  $101.98. 
(A-)  $421.89,  $13,641.11.     (Z)  $38.27,  $386.98. 

Exercise  83.— 1.  $21,  $35.     2.  $16.68,  $19.46.     3.  $53.82,  $47.84. 

4.  $23.12,  $28.90.       5.  $75,  $85.       6.  $128,  $119.       7.  $213.75,  $200. 
8.  $180,  $152.       9.  $120,  $175.        10.  $204.17,  $  163.33.       11.    $7.04, 
$8.60.        12.  $8.35,  $12.98.        13.  $9.39,  $6.26.        14.  $12.38,  $13.93. 

15.  $12.87,  $9.81.     16.  $16,  $30.     17.  $28.14,  $31.51.     18.  $9.19,  $4.92. 
19.  $12.85,  $13.49.  20.  $  56.07,  $56.07.  21.  $203.52,  $  126.64. 
22.  $243.83,  $190.           23.   $  164.79,  $105.94.  24.  $175.50,  $157.50. 

25.  $205,  $87.47.       26.  $191.25,  $  183.75.       27.  $16.11.        28.  $47.14. 
29.  $39.33.     30.  $53.36. 

Exercise  84.  — 1.  $7.?0.         2.  $12.18.         3.  $97.39.        4.  $100.07. 

5.  $74.19.     6.  $98.41.     7.  $64.51.      8.  $43.63.     9.  $16.46.     10.  $7.12. 
11.  $13.90.        12.  $117.26.        13.  $51.08.        14.  $8.46.        15.  $87.88. 

16.  $42.37. 


340  ANSWERS 

Exercise  85.  —  1.  $838.67.      2.  $693.10.      3.  $802.50.      4.  $365.61. 

5.  §337.79.        6.  §753.13.        7.  $880.28.        8.  $330.46.        9.  $1016.92. 
10.  $389.       11.  $1332.34.      12.   $3750.25.      13.  $1532.      14.   $4454.04. 
15.  $1243.63.       16.  $499.       17.  $826.40.       18.  $562.83.       19.  $667.55. 

20.  $880.10.     21.  $1014.30.     22.  $2518.56.     23.  $3935.15.     24.  $2706.41. 
25.  $1044.83.     26.  $1993.18.     27.  $960. 

Exercise  86.  —  1.  7830  oz.,  2890  oz.,  1200  oz.         2.  .579  oz.,  8.355  Ib. 

3.  77.78  Ib.  nearly.      4.  2.7.     5.  1.036.     6.  1.04.     7.  14.7241b.     8.  62.5T. 
9.  4.17  T.       10.  11.146  oz.,  6.059  oz.,  12.442  oz.,  6.586  oz.        11.  345  Ib. 
12.  1.0125  T.         13.  39|  Ib.         14.  54  T.         15.  810.86  Ib.         16.  1.3. 
17-  2.402  cu.  in.  18.  166J  Ib.  19.  280  oz.  20.  4.6955  times. 

21.  108.194  Ib.     22.  212.5  Ib. 

Exercises?.  — 1.  125ft.    2.  5^  yd.     3.  40  rd.    4.  14ft.     5.  1611  bu. 

6.  291  bu.      7.  19.36  bu.      8.  20  yd.      9.  17  yd.      10.  32  yd.      11.  10  ft. 
12.  4.5ft.     13.  7ft.     14.  44ft. 

Exercise  88.  — 2.  1792  cu.  ft.,  1433.6  bu.,  45,875.2  Ib.  3.  1725  cu.  ft., 
4800  turkeys.  4.  2720  cu.  ft.,  130,560  Ib.,  fff.  5.  20  cords,  $95,  $30. 

6.  2675.2  T. 

Exercise  89.— 2.  71,614^  Ib.  Troy,  58,928f  Ib.  Avoir.  3.  371.25  grains. 
4.51.050.  5.  36£  Ib.  Avoir,  3685 \  Ib.  Avoir.  6.  232.2  grains.  7.  $20.672. 
8.  61^.  9.  30.5. 

Exercise  90.  — 2.  $80,  $50.         2.  $591,  $394.         3.  $1440,  $1800. 

4.  $1188,  $2376,  $3564.       5.  $1700,  $2550,  $2975.       6.  756,  972,  1188. 

7.  912.5,  1095,  1277.5,  1460.     8.  1508,  1624,  1740,  1856.       9.  1176,  1232, 
1288,  1344.     10.  540,  612,  684,  756.     11.   57,  38.     12.  992,  744.      13.   140, 
225.     14.975,546.     15.1170,1300,1755.     16.396,770,1023.     17.  $490, 
$700,  $800.  18.  $ 720,  $ 900,  $  1080.  19.    $300,  $255,  $345. 

20.  $4444f,  $5555f. 

. Exercise  91.  —  1.   $8.25.        2.   11.2  da.        3.    $110.50.        4.   7J  da. 

5.  5f°.       6.    311  A.       7.    56ft.       8.    11J  da.       9.    $24.18.       10.    60  mi. 

11.  $185.      12.   $333.75.      13.   3960  ft.,  66  ft.      14.   52.8ft.     15.  4J  sec. 

16.  186,000  mi.  nearly.         17.   $112.50.         18.    12  J  yd. 

Exercise  92.— 1.    If        2.    15f.       3.    6J.       4.   The  first.       5.    3^. 

6.  0.        7.    50  Ib.        8.    ?\.        9.    4950  ft.        10.    .1212+  rd.        11.    .456. 

12.  .013875  T.  13.    1400  Ib.  14.    45°.  15.    12°.  16.    J. 

17.  .3,  $1500,  A's$750,  B's  $300.     18.   $1875.      19.    7.9.      20.    11.3984. 

21.  .0524.      22.   $1178.85.      23.    112.5.      24.    228.     25.    $150.      26.    J-i. 
27.    3.9.          28.   $18.70.          29.   $1500.          30.    13J  da.          31.    6  da. 
32.    9  da.      33.    14  hr.      34.   20  da.      35.    1.7875.      36.    JJ.      37.   325  Ib. 
38.  $326.67.      39.  $64,  $240.      40.  $75.     $313.32,  25  mo.     41.  $89.25, 


ANSWERS  341 

$107.10.  42.  $6.06.  43.  126  Ib.  15J  oz.  44.  $18.20.  45.  $121.67. 
46.  11J0,  $11.25.  47.  110,  $11.  48.  130,  $13.  49.  $34,588,  $32,858.60. 
50.  $133,016.  51.  $62.675.  52.  $117.60.  53.  5  mills,  500.  54.  21.82 
knots  per  hr.,  25.126  mi.  per  hr.  55.  20.584  knots  per  hr.,  23.703  mi. 
per  hr.  56.  24.13  knots  per  hr.  57.  65.5  mi.  nearly.  58.  63.617  mi. 
per  hr.  59.  50.77  mi.  per  hr.  60.  Wheat,  $1.31J;  corn,  81J0  ;  oats,  $1.06J 
nearly.  61.  (a)  $2367.55,  (b)  $6450.05,  (c)  $922.32,  (d)  $950.82, 
(e)  $4585.84,  (/)  $693.45,  (g)  $955.79. 

Exercise  93.— 1.    75,453,652.  2.    74,944,016.          3.   73,781,838. 

4.  80,963,637.  5.  80,903,198.  7.  New  England,  7607.2,  7619.39, 
7680.92  ;  Middle  Atlantic,  22,930.26,  23,149.83,  23,408.29 ;  Central 
Northern,  42,741.79,  43,251.64,  43,958.89;  South  Atlantic,  22,880.92, 
23,589.39,  24,179.92;  Gulf  and  Miss.  Valley,  17,567.13,  18,297.28, 
19,025.73;  Southwestern,  43,068.45,  44,852.05,  46,061.39.  8.  12T^. 

9.  llf.  10.  111.  11.  17|7.  12.  9T\.  13.  23£J.  14.  22}i. 
15.  30r78-.  16.  147°  24'  40".  17.  35  ft.  6  in.  18.  45  qt.  1  pt. 

19.  30  gal.  Iqt.       20.    41  pk.  3  qt.      21.   25  bu.  1  pk.      22.   50  hr.  18  min. 

23.  43  da.  2  hr.          24.   83  yd.  25.    156  times.  26.    27.63  liters. 
27.   23  ft.  8  in.     28.  4.37  mi. ;  13,431  carloads  ;  100.7325  mi.     29.  134  T. 

Exercise  94.  — 2.  199.98.  3.  398.63.  4.  $74.27.  5.  $697.16. 
6.  $98.25.  7.  $999.95.  8.  $223.81.  9.  $50.79.  10.  $310.16. 

11.  $36.65.      12.    99.99.      13.   3.2976.      14.   .9937.      15.   4.43.      16.   9.5. 
17.   5.4.     18.   45.01.     19.   2|.     20.   lOjf.     21.   5T^.      22.   9JJ.     23.   4-||. 

24.  2|J.     25.    5-ij.     26.   2J|.     27.    3?V     28.    11^.     29.    2JJ.     30.  11^. 
31.   10 .in.     32.  4  ft.  7  in.     33.  9  ft.  9  in.     34.  8  ft.  11  in.     35.    4  Ib.  11  oz. 
36.   11  Ib.  11  oz.      37.    14  Ib.  10  oz.      38.   2  hr.  46  min.      39.   55°  45'  45". 
40.   64°  55'  10".         41.    31°  26'.          42.    11  pk.  4  qt.          43.    9  pk.  7  qt. 
44.   17,357.     45.    12,219.     46.    5176.     47.    $8592.     48.    144,805  sq.    mi. 
49.    23,672,000,     78,272,000,     147,925,000.        50.    91,895.        51.    574  mi. 

Exercise  95.— 6.  27,878,400  ;  1,040,400.  7.  1,871,104  ;  6,921,495. 
8.  Corn,  $12.09;  wheat,  $10.34;  oats,  $9.89;  rye,  $9.84;  barley, 
$11.74;  buckwheat,  $11.09.  9.  30,759,200  Ib.  10.  N.H.,  $14.37; 
Mass.,  $22.27;  Conn.,  $17.55;  N.Y.,  $15.49;  N.J.,  $21.05;  Penn.,  $17.42; 
Md.,  $17.01;  Va.,  $19.37;  S.C.,  $22.26;  Ga.,  $25.99;  Ala.,  $25.93; 
La.,  $22.19;  Tenn.,  $20.31;  Ky.,  $17.89;  111.,  $12.25;  Minn.,  $9.35; 
Kan.,  $8.00;  Col.,  $23.75;  Utah,  $30.00;  Idaho,  $23.60;  Cal.,  $20.81. 

12.  (a)  $10.84,  (6)  $8.97,   (c)  $7.24,    (d)  $10.09,   (e)  $9.79,  (/)  $2.90, 
(g)  $56.53,  (A)  $55.68.      13.    97  ft.  J  in.      14.    73  ft.  9£  in.      15.    45  yd. 
1  ft. ;  67  ft.  8  in.         16.   46  Ib.  4  oz. ;  150  Ib.  3  oz.        17.   20  hr.  3  min. ; 
65  hr.  33  min.  18.    532°  ;  341°  1'  30''.  19.    29  pk.  2  qt. ;  59  pk. 

20.  49  gal.  2qt. ;   117  gal. 


342  ANSWERS 

Exercise  96.  — 1.  N.H.,  120.2  bu.,  72^;  R.I.,  125  bu.,  89^; 
Del.,  93  bu.,  69^;  N.C.,  77  bu.,  68^;  Fla.,  76  bu.,  120^;  Miss.,  110  bu., 
85  jZ;  W.  Va.,  88  bu.,  68^;  Mich.,  67  bu.,  56^;  111.,  75  bu.,  67^; 
Mo.,  82  bu.,  55^;  N.D.,  95  bu.,  38^;  Nev.,  120  bu.,  82?.  2.  .3048. 

3.  .3048.       4.    3.2808.       5.   1.6093.       6.    .6214.       7.  .4047.       8.  2.471. 
9.   .7646.      10.   1.3079.     11.   16J.     12.   21.     13.   32.     14.   21J.     15.   17J. 

16.  1£.      17.   If      18.   ||.     19.   If     20.   J.     22.   &.      23.   Jf.     24.   If. 
25.    jfo.      26.   45.      27.    2TV       28.    8J.      29.    3J.       30.   4f|.      31.   3f. 
32.   3  long  T. ;  5.905  long  T. ;  .00886  long  T. ;  3.9368  long  T.     33.   2.2046  T. ; 
7.7161  T. ;  .00551  T.        34.  6.0764  Ib.  Troy ;  16.076  Ib.  Troy  ;  8,037.69  Ib. 
Troy.  35.    5.76   Ib.  ;   11.023   Ib. ;  4.409   Ib.  36.    .0001894   mi. 
37.    .003125  mi.      38.    .00625  A.     39.   .0016625  sq  mi.     40.    .0002066  A. 
41.  .0005  T.    42.363.    43.  114$  barrels.    44.600.   45.  $6160.    46.  171.6  T. 

Exercise  98.  —  1.  24°  10'.  2.  14°  16'.  3.  11°  17' 40".  4.  29°  17'  46". 
5.  108°  63'  48".  6.  73°  28'.  7.  229°  22'  32".  8.  94°  60'  10". 
9.  116°  5'  40".  10.  88°  10'  64".  11.  114°  8'  45".  12.  78°  37'  37". 
13.  25°  33' 35".  14.  35°  23' 54".  15.  120°  36' 40".  16.  74°  44'  65". 

17.  116°  4- 2". 

Exercise  99  (answers  correct  to  the  second).  —  1.  1  hr.  42  min.  14  sec. 
2.  3  hr.  9  mm.  2  sec.  3.  4  hr.  35  min.  14  sec.  4.  12  hr.  53  min.  56  sec. 
5.  15  hr.  9  min.  38  sec.  6.  6  hr.  50  min.  69  sec.  7.  8  hr.  32  min. 
15  sec.  8.  6  hr.  33  min.  43  sec.  9.  15  hr.  20  min.  10  sec.  10.  13  hr. 
41  min.  28  sec.  11.  13  min.  38  sec.  12.  7  hr.  13  min.  58  sec.  13.  2  hr. 
10  min.  26  sec.  14.  6  hr.  46  min.  56  sec.  15.  14  hr.  13  min.  10  sec. 
16.  14  hr.  15  min.  2  sec.  17.  1  hr.  12  min.  69  sec. 

Exercise  100.  —  1.   13°  22'  6"  E.      2.   4°  20'  36"  E.      3.   88°  17'  36"  E. 

4.  3°  12'  24"  W.       5.   9°  66' 36"  E.      6.   1' 54"  W.       7.   3°  43' 64"  W. 
8.   16°  17'  36'   E.       9.   72°  30'  45".  W.      10.   2  hr.  37  min.  20  sec. 

Exercise  101.  — 1.  72°  W.  2.  11  o'clock  A.M.  March  3  ;  1  o'clock 
P.M.  March  3  ;  4  o'clock  A.M.  March  3.  3.  9  o'clock  A.M.  in  London, 

Manchester,  Glasgow  ;  5  o'clock  P.M.  in  Tien-Tsin  ;  11  o'clock  A.M.  in 
Constantinople.  4.  5  A.M.  5.  165°  W.  6.  6  A.M.  following  day. 
7.  6.30  P.M.  ;  9  P.M.  previous  day  ;  12.30  P.M.  8.  2  P.M.  ;  3  P.M.  ;  11  P.M.  ; 
11.30  P.M.  9.  4  hr.  50  min.  39  sec.  A.M.  ;  1  hr.  50  min.  39  sec.  P.M.  ; 

6  hr.  20  min.  39  sec.  P.M.  ;  11  hr.  20  min.  39  sec.  P.M.  10.  11  hr. 
30  min.  P.M.  previous  day ;  9  hr.  30  min.  P.M.  previous  day  ;  6  hr. 
30  min.  P.M.  previous  day.  11.  12.30  A.M.  following  day;  6.30  A.M. 
following  day. 

Exercise  102.— 1.    6  da.         2.   2£  hr.         3.   $592.20.         4.   25yd. 

5.  16  da.     6.   132°.     7.  21.6ft.     8.  600  mi.     9.   2355  mi.     10.  2030  mi. 


ANSWERS  343 

11.  605  mi.  12.  $51,937,925.  13.  $730,377.  14.  $2,944,492. 
15.  $.238.  16.  $.498.  17.  19.3^.  18.  1.09375. 

Exercise  106. —  1.  4.  2.  3  a.  3.  3  a.  4.  4.  5.  3  a2.  6.  3  a. 
7.  2  a2.  8.  6  a.  9.  3  a2.  10.  11  a3.  11.  11  a3.  12.  4  a2.  13.  2  a2. 
14.  3  a2.  15.  5  a2.  16.  7  x.  17.  3  a*.  18.  1  x.  19.  5x3.  20.  5x. 
21.  6.  22.  2z4.  23.  36.  24.  363.  25.  2  63. 

Exercise  108. —1.  9.  2.  8.  3.  20.  4.  30.  5.  18.  6.  35. 
7.  30.  8.  40.  9.  22.  10.  45.  11.  55.  12.  55.  13.  54. 

14.  60.        15.    45.       16.   49.       17.    90.       18.    55.         19.    63.        20.    40. 
21.    56.        22.    18.         23.    22.         24.    30.        25.    35. 

Exercise  109. —  1.  $675.  2.  $23.04.  3.  $609.20.  4.  10  da. 
5.  $2175.  6.  16  hr.  48  min.  7.  20  da.  8.  18f  bu.  9.  9  da. 
10.  25  da.  11.  400  men.  12.  21  da.  13.  18  turkeys.  14.  66^. 

15.  280,176ft.         16.    75ft.         17.    157.5ft.        18.    40  mi.  per  hr. 
Exercise  110.— 1.    84  A.         2.    150ft.         3.    $260.       4.    2057.5  T. 

5.  20  ft.  6.  7  da.  7.  12  da.  8.  6  da.  9.  7  hr.  12  min. 

10.  $113.40.  11.  $365.625.  12.  2J  da.  13.  8  da.  14.  17.85  T. 
15.  17£hr. 

Exercise  111.— 1.  A' s  share,  $ 800  ;  B's  share,  $1000;  C's  share, 
$1200.  2.  A's,  $600;  B's,  $1500.  3.  A's,  $90 ;  B's,  $81.  4.  A's, 
$900;  B's,  $700.  5.  A's,  $780;  B's,  $954  ;  C's,  $420.  6.  A,  $300; 
B,  $300.  7.  $700,  $750.  8.  3J  A.  9.  $12.50. 

Exercise  112.  — 1.  4%,  8%,  7}%,  6J%,  16 j%.  2.  .045;  .15;  .125; 
.625;  .0625;  .036.  3.  107.55;  18.335;  34.475;  .3594;  .62345. 

4.  315.72;  31.242;  .1596;  .18582;  .3846;  .04698.  5.  186.75;  63.81; 
320.4;  1.95435;  2.39265;  .1089.  6.  Land,  $19,466;  fencing, 

$2919.90;  earthworks,  $46,718.40;  tunnels,  $23,359.20;  viaducts  and 
bridges,  $33,092.20;  works,  $3893.20;  culverts,  $9733;  way, 
$22,385.90;  sidings,  $5839.80;  junctions,  $1946.60;  stations, 
$12,652.90;  legal  expenses,  $11,679.60;  maintenance,  $973.30.  7.  Ans. 
correct  to  one-tenth  of  one  million.  $974,500,000;  $54,200,000; 
$16,900,000;  $66,300,000;  $97,800,000;  $16,800,000.  9.  Ans.  correct 
to  one-tenth  of  one  million.  N.Y.,  $607,000,000  ;  Savannah,  $64,900,000; 
Boston,  $98,700,000;  Puget  Sound,  $49,200,000;  New  Orleans, 
$150,000,000;  Detroit,  $35,200,000  ;  Galveston,  $166,400,000 ;  Buffalo 
Creek,  $30,000,000;  Philadelphia,  $82,500,000;  Mobile,  $21,800,000; 
Baltimore,  $110,000,000;  Newport  News,  $20,100,000;  San  Francisco, 
$39,900,000;  Wilmington,  $18,500,000. 

Exercise  113.  -  Michigan,  4,725,000  Ib. ;  Minnesota,  1,176,000  Ib.  ; 
Alabama,  341,250  Ib.;  Montana,  12,535,250  Ib. ;  Wyoming,  10,511,920  Ib.  j 


344  ANSWERS 

Idaho,  5,578,650  Ib.  ;  California,  4,331,250  Ib. ;  Utah,  4,322,500  Ib.  ;  New 
Mexico,  6,061,000  Ib.  ;    Colorado,  3,118,500  Ib. ;   Arizona,  1,502,800  Ib.; 
Texas,  3,182,400  Ib. ;  Washington,  1,466,250  Ib. 
Exercise    114.  — 1.    1973.  2.  3990.          3.    $800.          4.    $1000. 

5.  $3004.44.         6.    $100.80.         7.    Maine,  $351,577,436.         S.Pennsyl- 
vania,    $3,910,701,678;          South     Carolina,     $195,620,105;     Kansas, 
$363,010,660;     Tennessee,    $405,641,915;     Washington,    $260,948,609; 
Texas,  $1,021,158,657. 

Exercise  115. —1.    Europe,  34.16%;  Asia,  8.99%;  Africa,  2.94%; 
North  America,   44.97%;    South  America,  5.84%;   Australasia,  3.10%. 

2.  Austria,    123.8%;   England,  7.5%;  France,  7.9%;  Germany,  4.4%; 
Ireland,   13.7%;     Scotland,   3.6%.       3.   63J%.        4.   56£%.         5.    8J%. 

6.  15^.       7.   20^perlb.       8.   4%.      9.    27J%.        10.    80^.       11.   $20. 

12.  17}%.         13.   100  Ib.         14.    9|%.         15.    16}%. 

Exercise     116.  —  1.    $54.60;     $782.60.          2.    $70.35;     $740.35. 

3.  $126;     $1386.          4.   $36.83;    $421.83.  5.    $137.50;    $2887.50. 
6.    $267.60;  $3612.60.          7.    $34.80;  $817.80.         8.   $41.79  ;  $638.79. 

9.  $56;  $3056.       10.    $37.60;  $977.60.       11.    $9;  $1809.       12.    $231; 
$2331.     13.    $47.97;   $1007.97.     14.    $189.56;    $3100.81.     15.    $136.18; 
$1993.18.     16.    $70.30;  $2845.30.     17.   $58.63 ;  $1828.63.     18.    $51.08; 
$2026.22.         19.    $43.73;  $1261.73.         20.    $  113.98  ;  $1901.98. 

Exercise  117.  —  1.    $3.88.        2.   $1.96.         3.    $1.18.        4.   $24.07. 
5.    $13.96.          6.    $26.02.          7.    $35.16.          8.    $49.79.          9.  $39.46. 

10.  $39.52. 

Exercise   118.  — 1.    $500.          2.    $1200.          3.    $1000.         4.    $240. 

5.  $2400.        6.   $300.        7.    $600.        8.    $800.        9.   $195.       10.   $450. 

11.  $1020.        12.    $324.         13.    $300. 

Exercise  119.  — 1.1  yr.         2.    3  yr.      3.    f  yr.      4.   4  yr.    5.   1  yr. 

6.  1  yr.  9  mo.          7.   3  yr.  3  mo.          8.    1  yr.  1  mo.  16  da.          9.    1  yr. 
10  mo.  15  da.        10.   1  yr.  1  mo.  15  da.        11.   1  yr.  10  mo.        12.   8  mo. 
15  da.         13.   2  yr.  11  mo. 

Exercise  120.  - 1.   6%.         2.   7%.        3.   6%.        4.   8%.        5.    7%. 
6.   6%.         7.    5%.         8.    4%.       9.  5%.     10.    4ff%.     11.    8%.      12.   9%. 

13.  4}%.          14.   6%.         15.   6%.         16.    7%.         17.   5%.         18.   6%. 

19.  5}%.        20.   4}%. 

Exercise    121.  — 1.   $800.           2.   $700.          3.    $600.  4.    $840. 

5.    $360.            6.    $230.            7.    $11,505.  8.    $162.50.  9.   $580. 

10.   $630.            11.   $740.            12.   $450.          13.   $403.  14.   $240. 

15.   $1917.50.        16.   $1800.         17.   $245.  18.   $240.  19.   $22,530. 

20.  $803.05. 


ANSWERS  345 

Exercise  122.  — 1.  $392.  2.  $6.43.  3.  $1.44.  4.  $5.51. 
5.  $24.57.  6.  $858.67.  7.  $548.91.  8.  $999.84.  9.  $14,000. 
10.  7%.  11.  7%.  12.  $24,000.  13.  $6000.  14.  1  mo.  18  da. 

15.  4  mo.      16.   2yr.  6  mo.      17.    1  yr.  8  mo.     18.   $48.      19.   $204.40. 
20.  219  da.     21.    2  mo.  12  da.     22.    $500.     23.   $1072.50.     24.   $13,000. 

Exercise  123. —1.  Discount  $4.13.  2.  Discount  $5.87.  3.  Dis- 
count $7.50.  4.  Discount  $7.00.  5.  Discount  $  9.50.  6.  Discount 
$1.00.  7.  Discount  $7.25.  8.  Discount  $2.25.  9.  Discount  $4.00. 

10.  Discount  $2.92.  11.   Discount  $13.33.  12.   Discount  $5.00. 

13.  Discount  $10.40.  14.   Discount  $3.37.          15.   Discount  $1.90. 

16.  Discount  $6.16.         17.   Discount  $  10.92.        18.    Discount  $  8.55. 

Exercise  124.  — 1.  Discount  01.65,  Proceeds  $352.72.  2.  Dis- 
count $9.25,  Proceeds  $391.87.  3.  Discount  $1.61,  Proceeds  $450.86. 

4.  Discount  $6.08,   Proceeds  $601.92.        5.   Discount  $5.05,  Proceeds 
$499.95.        6.   Discount  $6.64,  Proceeds  $898.31.       7.   Discount  $16.89, 
Proceeds  $996.44.      8.    Discount  $7.45,  Proceeds  $750.05.      9.   Discount 
$3.04,    Proceeds    $801.36.  10.   Discount    $4.24,  Proceeds  $399.96. 

11.  Discount  $21.38,  Proceeds  $843.49.        12.   Discount  $1.96,  Proceeds 
$904.79.          13.    Discount  $  17.60,  Proceeds  $  1201. 

Exercise  125.— 1.   $280.      2.   $720.13.        3.    $781.25.         4.   31$%. 

5.  20%,  $21.         6.   26%.        7.    (a)    $441,       (6)    $3528,      (c)  $2244, 
(d)  $4845,    (e)  $2565,    (/)$1344,    (gr)  2736,   (ft)  $1782,   (i)    $1642.20. 

Exercise  126. —  1.  $2835.13.  2.  $1776.30.  3.  $233.67. 

4.  $480.85.  5.  $1524.53.  6.  $479.46.  7.  $153.97.  8.  $427.89. 
9.  $268.02.  10.  $122.08. 

Exercise  127.  — 1.    $62.84.     2.    $215.47.     3.    $325.31.  4.  $197.25. 

Exercise    129.  —  1.   $2726.80.  2.    $3497.40.  3.    $5086.35. 

4.  $6305.72.  5.  $8279.64.  6.  $9955.05.  7.  $5484.76.  8.  $5287.92. 
9.  $6027.92.  10.  $6097.61.  11.  $5393.25.  12.  $9771.30.  13.  $4509. 

14.  $6913.08.  15.    $7789.72.  16.   $1232.46.  17.   $4.20. 
18.    $9987.50.         19.   $598.50.        20.   $993.33.        21.    $891. 

Exercise  130.  — 1.  25.215  francs,  20.45  marks,  23.973  crowns. 
2.  (a)  $86.85,  (6)  $106.15,  (c)  $75.04,  (d)  $214.20,  (e)  $93, 
(/)  $146.16,  (g)  $120.60.  3.  $5.  4.  2150.53  colons,  4926.11  crowns, 
3731.34  crowns.  5.  $38,600.  6.  $4980.  7.  .2055  libra.  8.  3.731 
crowns.  9.  $20.  10.  $10,800. 

Exercise    131.  — 2.    $582.94,     $618.49,    $725.14.  3.    $349.78. 

4.  $205.00.  5.  (a)  $40,270.80,  (6)  $11(J9.40,  (c)  $3792.  6.  (a)  $2654.75, 
(5)  $14,842.82, 


346  ANSWERS 

Exercise  132.  — 1.    $4872.               2.    $2671.14.  3.   $1116.50. 

4.    $17,064.12.                5.    $352.42.                6.    $1717.  7.    $289.50. 

8.  (a)  $4503.22,   (6)  $4134.75,   (c)  $600,   (<f)  $1798.45,  (e)  $1636.25, 
(/)  $2840.02. 

Exercise  133. —1.    £1200.  2.    £250.  3.    38,080  marks. 

4.  19,300  francs.  5.    2680  krones.  6.    £486  10s.  7.    5376.85 
francs.         8.    23,801.68  marks. 

Exercise  134.  — 2.    $1620,  $1640.  3.    $3720.  4.   $29,075. 

5.  $2965.62.         6.    $5337.50.      '7.    $8200  ;  $9750;  $9300 ;  $12,962.50  ; 
$14,287.50;  $13,412.50;  $10,850;  $14,412.50;  $16,837.50;  $6300. 

Exercise  135.  — 1.  $14,262.50  ;  $14,125.  2.  $9737.50.  3.  100 
shares.  4.  $125.  5.  $125.  6.  $400.  7.  200  shares.  8.  100 
shares.  9.  1000  shares.  10.  200  shares. 

Exercise  136.  — 1.  159}.  2.  149}.  3.  $18,550.  4.  $32,718.75. 
5.  4||o/o>  6  62a.  7.  $30,918.75.  9.  $875;  no  brokerage. 

10.   5f%.        11.   $200,  $5093.76.         13.    Central.         14.    438  shares. 

Exercise  137.  — 1.   $62.40.        2.   $500.         3.    $2.16.  4.   $26.28. 

5.  £43  4s.,  or  $210.23.            6.    $1.81.             7.   $1.46.  8.    $256.80. 

9.  $58.          10.   $72,  $8.19  each.          11.   $91.80.           12.  $17.50,  $81. 
13.  $65.       14.  $552,  $2.208.        15.  $750.        16.  $28.80.  17.  $94.50. 
18.    $125.         19.    £5.         20.    $82.12,29.4^. 

Exercise  141.  — 1.  .01,  .04,  .09,  .16,  .25,  .36,  .49,  .64,  .81.  2.  1,  J, 
i»  re?  A»  A»  A>  A»  8T>  Trioi  T?T>  ii?>  ri-g?  rirs>  si?»  siff»  *}?»  *ii»  *ii» 

^o-     3.  t,  A,  it,  A,  j&»  ifj,  m  ^v    4.  2j,  5|,  if i,  IA,  SA, 

3ft,  4fj,  39 A-         6.   1,  J,   sV,  ^,  Ti5,  ^  rfs^sh.   7l*   ioW»  TsVp 

T?V^'    ^1^7'    ^TT¥?    "SlVs'    4^^'    ¥"^r"J'    3FS^»    ^S^'    S'TjW'  ^'     3025. 

8.   (a)  3,    (6)  85.6735,    (c)  29.1708,    (d)  66.0806,  (e)  .4670,   (/)   .5790, 
(0)  .8991,   (ft)   .5,   (0   .6,   (j)   .9. 

Exercise  142.  — 1.    13.           2.   21.           3.   25.           4.    31.  5.    32. 

6.  43.         7.    53.         8.    62.         9.    65.         10.    73.         11.    76.  12.    82. 
13.  84.         14.  87.         15.  92.         16.  96.         17.  98.         18.  99.  19.  89. 
20.   78.        21.    69.        22.    69.        23.    49.        24.    58. 

Exercise  143.  —  1.  317.  2.332.  3.347.  4.414.  5.436. 
6.  447.  7.  479.  8.  527.  9.  557.  10.  595.  11.  626.  12.  676. 
13.  689.  14.  708.  15.  809.  16.  879.  17.  905.  18.  909. 

Exercise  144.  — 1.  .388.  2.  .496.  3.  .587.  4.  .539.  5.  .513. 
6.  .679.  7.  .729.  8.  .785.  9.  .885.  10.  .288.  11.  .249. 
12.  .0608, 


ANSWERS  347 

Exercise  145.  —  1.    1.095.  2.   2.062.  3.    1.049.  4.   2.28. 

5.  1.817.        6.  2.291.        7.  1.768.         8.  3.028.         9.  1.173.         10.  2.121. 

11.  1.696.        12.    .764.         13.    .816.         14.   .632.         15.    .7977. 
Exercise  146.  — 1.   69.57yd.        2.   241yd.        3.  238yd.        4.  1.025 

mi.        5.   739  rd.         6.    311.13  yd.,  155.56  yd.        7.   538.89  yd.,  179.63 
yd.         8.    19.41  rd. 
Exercise  147.  — 1.   10.          2.    13.          3.   17.          4.   29.          5.    10(3. 

6.  101.          7.    145.          8.    89.          9.    149.  10.    68.5.  11.   425. 

12.  305.  13.   433.  14.   305.  15.   50ft.  16.   14.14  rd. 

17.  30.232  rd. 

Exercise  148.— 1.  152.  2.  184.  3.  280.  4.  2.17.  5.  .2P1. 
6.  .319.  7.  .2.  8.  .748.  9.  .455. 

Exercise  149.  — 1.  126.  2.  180.  3.  264.  4.  840.  5.  522. 
6.  9240.  7.  150,769.  8.  8.625  A.  9.  5.775  A.  10.  1.638  A. 
11.  1.68  A.  12.  43.301  sq.  rd.  13.  1082.53  sq.  rd.  14.  182.25  sq.  in. 

15.  2592  sq.  ft.       16.  57.42  ft. 

Exercise  150.  — 1.  69.12.        2.   144.51.         3.  471.24.  4.   515.22. 

5.    615.75.           6.    420.97.           7.   540.35.            8.    22.62.  9.    37.07. 

10.  45.87          11.  102.         12.   152.        13.    7.8.        14.    13.27.  15.   60. 

16.  8.5.        17.  9.6.         18.    6.7.         19.   480  times. 

Exercise  151.  — 1.   615.8.  2.    1520.5.  3.   4071.5.  4.    69.4. 

5.  132.73.           6.   232.35.  7.    295.59.  8.   4778.4.  9.   3217. 
10.   7238.2.         11.    4300.8.  12.   6647.6.  13.    795.8.  14.    484.15. 
15.    548.2.          16.   688.3.  17.  6.023.  18.   3.789.  19.   7.643. 
20.   9.282. 

Exercise  152.  — 1.   31.        2.   17.         3.   33.  4.  42.  5.   2.6. 

6.  7.2.        7.  9.2.        8.   9.8.        9.  11.8.        10.  13.4.        11.  7.6.        12.  9.3. 

13.  99.         14.   850.        15.   650.         16.  39.25yd.          17.   288,576,452.4. 

18.  329.9  sq.  in.        19.    1661.9  sq.  in.         20.   259.8.         21.    139.1  sq.  in. 
22.  259.8  sq.  in.,  225  sq.  in.        23.    Circle. 

Exercise  153.— 1.  46.5  in.  2.  49.22  in.  3.  1.01ft.  4.  36°. 
5.  3°  36'.  6.  31' .2. 

Exercise  154.  — 1.  20ft.  2.  176ft.  3.  110  mi.  4.  7J  mi. 
5.  22.36  ft.  6.  7.07  ft.  7.  16  : 121.  8.  4:9.  9.  186.96  sq.  mi. 

Exercise  155.  — 1.  576  sq.  ft.  2.  680ft.  3.  lOf  ft.  4.  7J  ft. 
5.  2080  sq.  ft.,  $52.  6.  630  sq.  ft.  7.  $26.40.  8.  147  sq.  yd.  9.  53.45 
sq.  ft.  10.  4021  sq.  in.  11.  67  yd.  12.  2984.5  sq.  ft.  13.  15,456.6 
sq.  in.  14.  8392.70.  15.  120,687  sq.  in.  16.  5541.7  sq.  in. 

17.  28,842,700   sq.  mi.         18.    186,265,000  sq.    mi.         1'9.    (1)    Jupiter, 
23,235  million  sq.  mi. ;  (2)  Uranus,  3217  million  sq.  mi.  ;  (3)  Neptune, 


348  ANSWERS 

3421  million  sq.  mi. ;  (4)  Saturn,  16,741  million  sq.  mi.         20.   58  in. 
Exercise  156.  — 1.   5280  cu.  in.         2.    700  cu.  in.        3.    4071. 5  cu.  in. 

4.  15,708  cu.  in.        5.    2598  cu.  in.         6.   33,510  cu.  in.         7.   2144.7 
cu.  ft.  8.    4094  gal.  9.   4562  gal.  10.    3:2.  11.    1:2. 
12.    52  cu.  ft.  1269  cu.  in.         13.    3  ft.         14.    4189  cu.  in.         15.   40  ft. 
16.    1,367,631.        17.   48  times.         18.    760  times.         19.   1331  times. 

Exercise  157. —1.   30°  C.         2.   25°  C.          3.   95°  C.         4.    120°  C. 

5.  20°  C.          6.    12f°C.          7.   3i°C.  8.    -  5°  C.          9.    -  10°  C. 

10.  -25°C.  11.    -40°C.  12.    -67J°C.  13.    95°  Fahr. 
14.   131°  Fahr.          15.   77°  Fahr.          16.   68°  Fahr.           17.    64.4°  Fahr 
18.   46.4°  Fahr.         19.    14°  Fahr.         20.    -  4°  Fahr.         21.    6.8°  Fahr. 
22.  0.4°  Fahr.        23.   -  11.2°  Fahr.         24.  -  459.4°  Fahr.         25.  Mer- 
cury,   -40°;   sulphur,    235.4°;    lead,    618.8°;    zinc,    779°;   gold,    1895°; 
cast  iron,  2012°  to  2102°.        26.    39.2°. 

Exercise  158.  — 1.    3,921,138.813  meters.  2.    107,934,859.86cm. 

3.  3,131,587.7mm.         4.   28.434km.        5.    19454m.         6.    140.784m. 
7.    960  times.  8.    7.03  times.  9.   5000  times.  10.    112|. 

11.  20,000.         12.   62.5  times.        13.   5000.         14.    1200,  480. 
Exercise  159.  — 1.   7.8125  ha.  2.    1.8605  ha.  3.    5.7122  ha. 

4.  6.82  ha.         5.   11.95  ha.        6.    6.65  ha.         7.    .81  cbm.         8.   3.36  a. 
9.  1.91  ca.     10.  24,429  c.cm.     11.  119  cm.     12.  1596  cbm.     13.  169.65  ca. 
14.   51.08  km.       15.  462.3  qcm. 

Exercise  160.  — 1.  $1156.43.  2.  |622.91.  3.  $1650. 

4.  $1278.12.  5.  $983.59. 

Exercise  161.  — 1.  $2737.14.  2.  $4031.75.  3.  $8005.16. 

4.  $9773.37.  5.  $11,876.05.  6.  $2853.54. 

Exercise  162.  — 3.  2J  hr.  4.  IJf  hr.  5.  4T8g  hr.  6.  £J. 

7.  f  hr.  8.  1  hr. 

Exercise  163.  — 1.  1260  mi.  2.  5  hr.,  150  mi.  3.  10.38A.M. 

4.  22  mi.  per  hr.  5.  105.6  yd.  6.  180  mi. 

Exercise  164.  —  1.  48,  72,  etc.  2.  6  min.  spaces.  3.  lO^min. 
past  2  o'clock,  16^  min.  past  3  o'clock,  27T3T  min.  past  5  o'clock,  38r2T 
min.  past  7  o'clock,  49^  min.  past  9  o'clock,  54T6T  min.  past  10  o'clock. 
At  no  time.  5.  (a)  38T2T  min.  past  1  o'clock,  (&)  49Jr  min.  past  3 
o'clock,  (c)  10}£  min.  past  8  o'clock,  (d)  16^  min.  past  9  o'clock, 
(e)  27T3r  min.  past  11  o'clock,  (/)  32T8T  min.  past  12  o'clock.  6.  49^ 
min.  past  3  o'clock,  5j\  min.  past  7  o'clock.  7.  (a)  27^  min.  past 
2  o'clock,  (&)  32T*r  min.  past  3  o'clock,  (c)  5j5r  min.  past  4  o'clock, 
38r2T  min.  past  4  o'clock,  (d)  16r4T  min.  past  6  o'clock,  49T11  min.  past 
0  o'clock,  (e)  27^  min.  past  8  o'clock,  (/)  32T8r  min.  past  9  o'clock, 


ANSWERS  349 

ft)  16T4r  min.  past  12  o'clock.  8.  13^- 
min.  past  4  o'clock,  5  o'clock,  48  min.  past  4  o'clock.  9.  38T2r  min. 
past  5  o'clock. 

MISCELLANEOUS   EXAMPLES   (A) 

8.  8999.991.  9.  274.999225.  10.  8,447,537,940,492.  11.  636,300,000. 
12.  751,700,800.  13.  7.1407.  14.  814,585.36.  15.  166.375  mi. 
16.  $3828.12.  17.  $3670.12.  18.  .000504.  19.  1.12550881. 

20.   278,500,000.  21.    (1)   $.475,     (2)    $3.508,      (3)    $51.877. 

22.    372.015.  23.    28,127,000  nearly.  25.    52,800  mi. 

26.  .341.  27.  .8251.  28.  .1704.  29.  .000439625.  30.  14.461°. 
81.  8°  39' 6".  32.  80.7  X'.  33.  113,400.  34.  250.  35.  80,032,000  nearly. 
36.  77.  37.  $3.78.  38.  $1800.  39.  $384.45.  40.  $30.80. 
41.  12,290.  42.  $20.25.  43.  .5774.  44.  .037.  45.  T|T,  or  .0390625. 
46.  144.  47.  2,  3,  6,  9,  18.  48.  2,  3,  4,  6,  8,  9,  12,  18,  24,  36,  72. 
49.  1110  =  2  •  3  •  5  .  37  ;  777  =  3  .  7  .  37  ;  1001  =  7  •  11  •  13  ;  L.C.M.  2  .  3  - 
5  -  7  •  11  -  37.  50.  26.  51.  65,520.  52.  25.  53.  19  ;  66,880. 
54.  2,  3,  52,  72,  11.  55.  2,784,873.  56.  2*  .38  -  11  .  13.  57.  25  A. 
58.  80  poles.  59.  },  f ,  JJ.  60.  fa.  61.  14  j}  62.  T%. 

63.  1.09375,  .109375,  .128.  64.  J>ff,  ^T,  ^.  65.  3&.  66.  1^. 
67.  .025.  68.  .027.  69.  .075.  70.  .16583.  72.  Upwards  of 
80,000  yr.  73.  (1)39.534,  (2)  48.321,  (3)  54.197,  (4)  63.25, 
(5)  63.615,  (6)  68.234,  (7)  72.932,  (8)  76.459,  (9)  79.68, 
(10)  82.951,  (11)  70.13.  74.  $11,687.50.  75.  20,000  bu. 

76.  $3062.50.  77.  $42.40.  78.  $92.67.  79.  $2.174.  80.  $728.42. 
81.  36.3  nearly.  82.  5,201,300.  83.  52,100,000  nearly.  84.  56,310,000. 
85.  12.44496  ft.  86.  712.5  Ib.  87.  320.  88.  1.292  sec. 

89.  4888fcu.  yd.  90.  .002055  nearly.  91.  $6256.  92.  $3344. 
93.  .2565.  94.  19s.  2J<2.  95.  .15625.  96.  (a)  147,824,  (ft)  22,098, 
(c)  463,180,  (d)  823,750.  97.  (a)  166,488,  (ft)  526.380.  98.  640  A. 

99.    2640  revolutions.  100.   112.5  bu.         101.    ^.  102.  $75. 

103.   $24,800.         104.    10  A.       105.   $  14f       106.   (i)  .00},    (ii)   .00375. 

107.    (a)    — ,  (ft)   — ,   (c)  J?t,   (d)    —  .•      108.   99.99.       109.    -2r6/. 
100         '   100        '   100  100 

110.  $107.         111.    2.28.         112.    iJ.         113.    66,720.         114.   $101.02. 

115.  25ft.        116.    35.         117.   847.15625.         118.    28J.         119.   $7260. 

120.  390.        121.   1092  eggs.         122.    ^f.          123.   23f-.          124.    17.66. 

125.  5.25ft.,  212. 0045  sq.  ft.,  98.328  sq.  ft.,    59.427  sq.  ft.       126.    $330. 

127.  67^  mi.  128.  7.15p.M.,  Apr.  5.  129.  9.55A.M.  130.  7  hr. 
29  min.  46  sec.  131.  63°  35'  W.  132.  156°  15'  W.  133.  135°  27'  E. 

134.  (a)  70°  15'  W.,       (ft)  85°  W.,       (c)  81°  45'  W.             135.   24.62%. 


350  ANSWERS 

136.  (i)  4.16%,  (ii)  14.6%,  (iii)  28.79%,  (iv)  17.8%.  137.  96,600. 
138.  $150.  139.  $1.44.  140.  20%,  or  16?  per  yard.  141.  $8. 
142.  24f%.  143.  }.  144.  $150.  145.  38|%.  146.  12J%. 

147.  (i)64TV/0,  (ii)3fgal.  148.  (a)  59.99%,  (6)32.34%,  (c)  62.84%, 
(d)  49.02%,  (e)  65.57%,  (/)  36.75%,  (?)  65.28%,  (ft)  40.83%, 
(0  41.94%,  0)  39.01%,  (fc)  42.02%.  149.  $1.61.  150.  $1.20- 

151.  $182.67.  152.  £7  17s.  8d  153.  £41  16s.  Id.  154.  $300. 
155.  $14,400.  156.  $400.  157.  $1200.  158.  $1000.  159.  7%. 
160.  $500.  161.  62^.  162.  $546.69.  163.  July  23,  $597.14. 

164.  $149.15.  165.  $891.80.  166.  $1004.72.  167.  Mar.  11,  1903, 
$824.89.  168.  $459.50.  '  169.  7500  sq.  yd.  170.  3500  sq.  yd. 

171.  $62.80.  172.  301|sq.  yd.  173.  1  mi.  174.  1  mi.  long,  J  mi. 
wide.  175.  4330  sq.  ft.  176.  6495  sq.  ft.  177.  355  in.  178.  314.16 
sq.  in.  179.  1256.64  sq.  in.  180.  1.9531  cu.  ft.  181.  6.07  in. 

182.  $.525.  183.  $.882.  184.  $1.  185.  $.612.  186.  $.756. 
187.  $.871.  188.  $1.802.  189.  $.755.  190.  $3.  191.  512.12  francs. 

MISCELLANEOUS  EXAMPLES  (B) 

1.   T|o.  2.    38°  15'.  3.    216.  4.    2.77ft.          5.    2.0575  yr. 

6.  3  gal.  1.02  gi.  7.  $4176.  8.  $20.94.  9.  Ill  bu.  1  pk.  2J  qt. 
10.  17f .  Ib.  11.  IfjjJ.  12.  $504.  13.  1.2857142.  14.  1.777+. 
15.  16ft.  16.  12.56A.M.  17.  $500.  18.  $11.28.  19.  3J  da. 
20.  2ii|.  21.  45J  bu.  22.  82.28  yr.  23.  23^.  24.  226.27+  rd. 
25.  $6.66.  26.  149.61+ gal.  27.  734  rd.  28.  90yd.  29.  $16.67. 
30.  137  da.  31.  $331.86.  32.  4531  mi.  33.  Answers  will  depend 
on  date.  34.  14.  35.  $9.60.  36.  10  A.  37.  30  inin.  50  sec. 
38.  6ch.  89.6+b.  39.  §116.35$.  40.  $37.50,  $30,  $52.50. 

41.  62.72  bu.  42.  3403J  T.  43.  Neither  gained  nor  lost.  44.  9r%. 
45.  2295.  46.  140  A.  47.  4844.996.  48.  $75.15,  $83.50,  $300.60. 
49.  6  hr.  1  min.  28  sec.  A.M.  50.  Answer  will  depend  on  day  calculation 
is  made.  51.  $2800.  52.  $604.86.  53.  $50  loss.  54.  25%. 
55.  $460.  56.  10.8%.  57.  16  yr.  8  mo.,  12  yr.  6  mo.,  11  yr. 

58.   $453.561.  59.    $352.35.  60.    4J-%.  61.   210,526.3  Ib. 

62.  8275.80.  63.  35J%.  64.  410  on  the  $  100,  $4.92.  65.  $6.67. 
66.  $56.25.  67.  $30,563.38+.  68.  80.  69.  2X\%.  70.  33J%. 
71.  $855.  72.  $27,200.  73.  $32.32  loss.  74.  $1092.56. 

78.   30%.  79.   1  yr.  7  mo.  12  da.  80.    $492.50.  81.    300%. 

82.  $3200.  83.  33  J%.  84.  35%.  85.  $37.525.  86.  $108.90. 
87.  25%.  88.  $753.  89.  $5.69.  90.  Dec.  13,  1902.  91.  $6644.63. 
92.  83515.63.  93.  3|%.  94.  $530.  95.  $15,000.  96.  $1533.75. 
97.  9f%  98.  Yes,  $20  better.  99.  $2042.  100.  25%. 

101.    $1000.         102.    $7045.          103.    $28.01,  $42.02.         104.    6}Jf 


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